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ates with angular velocity ω about the center of mass, then Since rc i = ri − R, we have i = ω × rc ˙rc i. ˙ri = ˙R + ω × rc i = ˙R + ω × (ri − R). On the other hand, the kinetic energy, as calculated in previous lectures, is T = 1 2 M \| ˙R\|2 + 1 2 mi\|˙rc i \|2 = 1 2 M \| ˙R\|2 translational KE i + 1 I cω2 2 rotational KE. Sometimes we do not want to use the center of mass as the center. For example, if an item is held at the edge and spun around, we’d like to study the motion about the point at which the item is held, and not the center of mass. So consider any point Q, with position vector Q(t) that is not the center of mass but moves with the rigid body, i.e. ˙Q = ˙R + ω × (Q − R). Usually this is a point inside the object itself, but we do not assume that in our calculation. Then we can write ˙ri = ˙R + ω × (ri − R) = ˙Q − ω × (Q − R) + ω × (ri − R) = ˙Q + ω × (ri − Q). Therefore the motion can be considered as a translation of Q (with diﬀerent velocity than the center of mass), together with rotation about Q (with the same angular velocity ω). Equations of motion As shown previously, the linear and angular momenta evolve according to ˙P = F (total external force) ˙L = G (total external torque) These two equations determine the translational and rotational motion of a rigid body. 55 7 Rigid bodies IA Dynamics and Relativity L and G depend on the choice of origin, which could be any point that is ﬁxed in an inertial frame. More surprisingly, it can also be applied to the center of mass, even if this is accelerated: If then mi¨ri = Fi, mi¨rc i = Fi + mi ¨R. So there is a ﬁctitious force mi ¨R in the center-of-mass frame. But the
total torque of the ﬁctitious forces about the center of mass is rc i × −mi ¨R = − mirc i × ¨R = 0 × ˙R = 0. i So we can still apply the above two equations. In summary, the laws of motion apply in any inertial frame, or the center of mass (possibly non-inertial) frame. Motion in a uniform gravitational ﬁeld In a uniform gravitational ﬁeld g, the total gravitational force and torque are the same as those that would act on a single particle of mass M located at the center of mass (which is also the center of gravity): F = Fext i = mig = M g, i i and G = i Gext i = i ri × (mig) = miri × g = M R × g. In particular, the gravitational torque about the center of mass vanishes: Gc = 0. We obtain a similar result for gravitational potential energy. The gravitational potential in a uniform g is Φg = −r · g. (since g = −∇Φg by deﬁnition) So V ext = = i i V ext i mi(−ri · g) = M (−R · g). Example (Thrown stick). Suppose we throw a symmetrical stick. So the center of mass is the actual center. Then the center of the stick moves in a parabola. Meanwhile, the stick rotates with constant angular velocity about its center due to the absence of torque. Example. Swinging bar. 56 7 Rigid bodies IA Dynamics and Relativity M g This is an example of a compound pendulum. Consider the bar to be rotating about the pivot (and not translating). Its 3 M 2. The gravitational torque about angular momentum is L = I ˙θ with I = 1 the pivot is G = −M g 2 sin θ. The equation of motion is So ˙L = G. I ¨θ = −M g 2 sin θ, or 3g 2 which is exactly equivalent to a simple pendulum of length 2/3, with angular ¨θ = − sin θ. frequency 3g 2. This can also be obtained from an energy argument: E = T + V = I ˙θ2 − M g 1 2 2 cos θ. We diﬀerentiate to obtain dE
dt = ˙θ(I ¨θ + M g 2 sin θ) = 0. I ¨θ = −M g 2 sin θ. So Sliding versus rolling Consider a cylinder or sphere of radius a, moving along a stationary horizontal surface. a ω v 57 7 Rigid bodies IA Dynamics and Relativity In general, the motion consists of a translation of the center of mass (with velocity v) plus a rotation about the center of mass (with angular velocity ω). The horizontal velocity at the point of contact is vslip = v − aω. For a pure sliding motion, v = 0 and ω = 0, in which case v − aω = 0: the point of contact moves relative to the surface and kinetic friction may occur. For a pure rolling motion, v = 0 and ω = 0 such that v − aω = 0: the point of contact is stationary. This is the no-slip condition. The rolling body can alternatively be considered to be rotating instantaneously about the point of contact (with angular velocity ω) and not translating. Example (Rolling down hill). a ω α v Consider a cylinder or sphere of mass M and radius a rolling down a rough plane inclined at angle α. The no-slip (rolling) condition is The kinetic energy is v − aω = 0. The total energy is T = 1 2 M v2 + 1 2 Iω2 = 1 2 M + I a2 v2. E = M + I a2 1 2 ˙x2 − M gx sin α, where x is the distance down slope. While there is a frictional force, the instantaneous velocity is 0, and no work is done. So energy is conserved, and we have dE dt I a2 = ˙x M + ¨x − M g sin α = 0. So M + I a2 ¨x = M g sin α. For example, if we have a uniform solid cylinder, I = 1 2 M a2 (as for a disc) 58 7 Rigid bodies IA Dynamics and Relativity and so For a thin cylindrical shell, So ¨x = 2 3 g sin α. I = M a2. ¨x = 1 2 g sin α. Alternatively, we may do it in terms of forces and torques, N F v M g α The equations of motion are and While rolling
, So M ˙v = M g sin α − F I ˙ω = aF. ˙v − a ˙ω = 0. M ˙v = M g sin α − I a2 ˙v, leading to the same result. Note that even though there is a frictional force, it does no work, since vslip = 0. So energy is still conserved. Example (Snooker ball). a M g F ω N 59 7 Rigid bodies IA Dynamics and Relativity It is struck centrally so as to initiate translation, but not rotation. Sliding occurs initially. Intuitively, we think it will start to roll, and we’ll see that’s the case. The constant frictional force is F = µkN = µkM g, which applies while v − aω > 0. The moment of inertia about the center of mass is The equations of motion are I = 2 5 M a2. M ˙v = −F I ˙ω = aF Initially, v = v0 and ω = 0. Then the solution is v = v0 − µkgt µkg a ω = 5 2 t as long as v − aω > 0. The slip velocity is vslip = v − aω = v0 − µkgt = v0 1 − 7 2, t troll where troll = 2v0 7µkg. This is valid up till t = troll. Then the slip velocity is 0, rolling begins and friction ceases. At this point, v = aω = 5 7 v0. The energy is then 5 14 M v2 0 < 1 2 M v2 0. So energy is lost to friction. 60 8 Special relativity IA Dynamics and Relativity 8 Special relativity When particles move Extremely FastTM, Newtonian Dynamics becomes inaccurate and is replaced by Einstein’s Special Theory of Relativity (1905). Its eﬀects are noticeable only when particles approach to the speed of light, c = 299 792 458 m s−1 ≈ 3 × 108 m s−1 This is really fast. The Special Theory of Relativity rests on the following postulate: The laws of physics are the same in all inertial frames This is the principle of relativity familiar to Galileo. Galilean relativity mentioned in the ﬁrst chapter satisﬁes this postulate for dynamics.
People then thought that Galilean relativity is what the world obeys. However, it turns out that there is a whole family of solutions that satisfy the postulate (for dynamics), and Galilean relativity is just one of them. This is not a problem (yet), since Galilean relativity seems so intuitive, and we might as well take it to be the true one. However, it turns out that solving Maxwell’s equations of electromagnetism gives an explicit value of the speed of light, c. This is independent of the frame of reference. So the speed of light must be the same in every inertial frame. This is not compatible with Galilean relativity. Consider the two inertial frames S and S, moving with relative velocity v. Then if light has velocity c in S, then Galilean relativity predicts it has velocity c − v in S, which is wrong. Therefore, we need to ﬁnd a diﬀerent solution to the principle of relativity that preserves the speed of light. 8.1 The Lorentz transformation Consider again inertial frames S and S whose origins coincide at t = t = 0. For now, neglect the y and z directions, and consider the relationship between (x, t) and (x, t). The general form is x = f (x, t), t = g(x, t), for some functions f and g. This is not very helpful. In any inertial frame, a free particle moves with constant velocity. So straight lines in (x, t) must map into straight lines in (x, t). Therefore the relationship must be linear. Given that the origins of S and S coincide at t = t = 0, and S moves with velocity v relative to S, we know that the line x = vt must map into x = 0. Combining these two information, the transformation must be of the form x = γ(x − vt), (1) for some factor γ that may depend on \|v\| (not v itself. We can use symmetry arguments to show that γ should take the same value for velocities v and −v). Note that Galilean transformation is compatible with this – just take γ to be always 1. 61 8 Special relativity IA Dynamics and Relativity Now reverse the roles of the frames. From the perspective S, S moves with velocity −v. A similar argument leads to x = γ(x + vt), (2) with the
same factor γ, since γ only depends on \|v\|. Now consider a light ray (or photon) passing through the origin x = x = 0 at t = t = 0. Its trajectory in S is We want a γ such that the trajectory in S is x = ct. x = ct as well, so that the speed of light is the same in each frame. Substitute these into (1) and (2) ct = γ(c − v)t ct = γ(c + v)t Multiply the two equations together and divide by tt to obtain So c2 = γ2(c2 − v2). γ = c2 c2 − v2 = 1 1 − (v/c)2. Deﬁnition (Lorentz factor). The Lorentz factor is γ = 1 1 − (v/c)2. Note that – γ ≥ 1 and is an increasing function of \|v\|. – When v c, then γ ≈ 1, and we recover the Galilean transformation. – When \|v\| → c, then γ → ∞. – If \|v\| ≥ c, then γ is imaginary, which is physically impossible (or at least weird ). – If we take c → ∞, then γ = 1. So Galilean transformation is the transformation we will have if light is inﬁnitely fast. Alternatively, in the world of Special Relativity, the speed of light is “inﬁnitely fast”. γ v c 1 62 8 Special relativity IA Dynamics and Relativity For the sense of scale, we have the following values of γ at diﬀerent speeds: – γ = 2 when v = 0.866c. – γ = 10 when v = 0.9949. – γ = 20 when v = 0.999c. We still have to solve for the relation between t and t. Eliminate x between (1) and (2) to obtain x = γ(γ(x − vt) + vt). t = γt − (1 − γ−2) γx v t − = γ v c2 x. So So we have Law (Principle of Special Relativity). Let S and S be inertial frames, moving at the relative velocity of v. Then
x = γ(x − vt) v c2 x t = γ t − where γ = 1 1 − (v/c)2,. This is the Lorentz transformations in the standard conﬁguration (in one spatial dimension). The above is the form the Lorentz transformation is usually written, and is convenient for actual calculations. However, this lacks symmetry between space and time. To display the symmetry, one approach is to use units such that c = 1. Then we have x = γ(x − vt), t = γ(t − vx). Alternatively, if we want to keep our c’s, instead of comparing x and t, which have diﬀerent units, we can compare x and ct. Then we have x = γ ct = γ x − ct − (ct), x. v c v c Symmetries aside, to express x, t in terms of x, t, we can invert this linear mapping to ﬁnd (after some algebra) x = γ(x + vt) t = γ t + v c2 x Directions perpendicular to the relative motion of the frames are unaﬀected: y = y z = z 63 8 Special relativity IA Dynamics and Relativity Now we check that the speed of light is really invariant: For a light ray travelling in the x direction in S: x = ct, y = 0, z = 0. x t = γ(x − vt) γ(t − vx/c2) = (c − v)t (1 − v/c)t = c, In S, we have as required. For a light ray travelling in the Y direction in S, In S, and and x = 0, y = ct, z = 0. x t = γ(x − vt) γ(t − vx/c2) = −v, y t = y γ(t − vx/c2 = c γ, z = 0. So the speed of light is x2 + y2 t = v2 + γ−2c2 = c, as required. More generally, the Lorentz transformation implies c2t2 − r2 = c2t2 − x2 − y2 − z2 2 t − = c2γ2 =
γ2 1 − v c2 x v2 c2 − γ2(x − vt)2 − y2 − z2 (c2t2 − x2) − y2 − z2 = c2t2 − x2 − y2 − z2 = c2t2 − r2. We say that the quantity c2t2 − x2 − y2 − z2 is Lorentz-invariant. So if r t = c, then r t = c also. 8.2 Spacetime diagrams It is often helpful to plot out what is happening on a diagram. We plot them on a graph, where the position x is on the horizontal axis and the time ct is on the vertical axis. We use ct instead of t so that the dimensions make sense. 64 8 Special relativity IA Dynamics and Relativity ct world line P x Deﬁnition (Spacetime). The union of space and time in special relativity is called Minkowski spacetime. Each point P represents an event, labelled by coordinates (ct, x) (note the order!). A particle traces out a world line in spacetime, which is straight if the particle moves uniformly. Light rays moving in the x direction have world lines inclined at 45◦. light ray ct light ray x We can also draw the axes of S, moving in the x direction at velocity v relative to S. The ct axis corresponds to x = 0, i.e. x = vt. The x axis corresponds to t = 0, i.e. t = vx/c2. ct ct x x Note that the x and ct axes are not orthogonal, but are symmetrical about the diagonal (dashed line). So they agree on where the world line of a light ray should lie on. 8.3 Relativistic physics Now we can look at all sorts of relativistic weirdness! 65 8 Special relativity IA Dynamics and Relativity Simultaneity The ﬁrst relativistic weirdness is that diﬀerent frames disagree on whether two evens are simultaneous Deﬁnition (Simultaneous events). We say two events P1 and P2 are simultaneous in the frame S if t1 = t2. They are represented in the following spacetime diagram by horizontal dashed lines. However, events that are simultaneous in S have equal values of t, and so
lie on lines ct − v c x = constant. ct ct P1 P2 x x The lines of simultaneity of S and those of S are diﬀerent, and events simultaneous in S need not be simultaneous in S. So simultaneity is relative. S thinks P1 and P2 happened at the same time, while S thinks P2 happens ﬁrst. Note that this is genuine disagreement. It is not due to eﬀects like, it takes time for the light conveying the information to diﬀerent observers. Our account above already takes that into account (since the whole discussion does not involve speciﬁc observers). Causality Although diﬀerent people may disagree on the temporal order of events, the consistent ordering of cause and eﬀect can be ensured. Since things can only travel at at most the speed of light, P cannot aﬀect R if R happens a millisecond after P but is at millions of galaxies away. We can draw a light cone that denotes the regions in which things can be inﬂuenced by P. These are the regions of space-time light (or any other particle) can possibly travel to. P can only inﬂuence events within its future light cone, and be inﬂuenced by events within its past light cone. 66 8 Special relativity IA Dynamics and Relativity ct Q P R x All observers agree that Q occurs after P. Diﬀerent observers may disagree on the temporal ordering of P and R. However, since nothing can travel faster than light, P and R cannot inﬂuence each other. Since everyone agrees on how fast light travels, they also agree on the light cones, and hence causality. So philosophers are happy. Time dilation Suppose we have a clock that is stationary in S (which travels at constant velocity v with respect to inertial frame S) ticks at constant intervals ∆t. What is the interval between ticks in S? Lorentz transformation gives t = γ t + v c2 x. Since x = constant for the clock, we have ∆t = γ∆t > ∆t. So the interval measured in S is greater! So moving clocks run slowly. A non-mathematical explanation comes from Feynman (not lectured): Suppose we have a very simple clock: We send a
light beam towards a mirror, and wait for it to reﬂect back. When the clock detects the reﬂected light, it ticks, and then sends the next light beam. Then the interval between two ticks is the distance 2d divided by the speed of light. d From the point of view of an observer moving downwards, by the time light reaches the right mirror, it would have moved down a bit. So S sees a d 67 8 Special relativity IA Dynamics and Relativity However, the distance travelled by the light beam is now (2d)2 + a2 > 2d. Since they agree on the speed of light, it must have taken longer for the clock to receive the reﬂected light in S. So the interval between ticks are longer. By the principle of relativity, all clocks must measure the same time dilation, or else we can compare the two clocks and know if we are “moving”. This is famously evidenced by muons. Their half-life is around 2 microseconds (i.e. on average they decay to something else after around 2 microseconds). They are created when cosmic rays bombard the atmosphere. However, even if they travel at the speed of light, 2 microseconds only allows it to travel 600 m, certainly not suﬃcient to reach the surface of Earth. However, we observe lots of muons on Earth. This is because muons are travelling so fast that their clocks run really slowly. The twin paradox Consider two twins: Luke and Leia. Luke stays at home. Leia travels at a constant speed v to a distant planet P, turns around, and returns at the same speed. In Luke’s frame of reference, ct 2cT R cT Luke A (Leia’s arrival) Leia: x = vt x Leia’s arrival (A) at P has coordinates (ct, x) = (cT, vT ). The time experienced by Leia on her outward journey is T = γ T − v c2 T = T γ. By Leia’s return R, Luke has aged by 2T, but Leia has aged by 2T she is younger than Luke, because of time dilation. γ < 2T. So The paradox is: From Leia’s perspective, Luke travelled away from her at speed and the returned, so he should be younger than her! Why is the problem not symmetric? We can draw
Leia’s initial frame of reference in dashed lines: 68 8 Special relativity IA Dynamics and Relativity ct R ct Z X Han A Leia x x In Leia’s frame, by the time she arrives at A, she has experienced a time T = T γ as shown above. This event is simultaneous with event X in Leia’s frame. Then in Luke’s frame, the coordinates of X are (ct, x) = cT γ, 0 = cT γ2, 0, obtained through calculations similar to that above. So Leia thinks Luke has aged less by a factor of 1/γ2. At this stage, the problem is symmetric, and Luke also thinks Leia has aged less by a factor of 1/γ2. Things change when Leia turns around and changes frame of reference. To understand this better, suppose Leia meets a friend, Han, who is just leaving P at speed v. On his journey back, Han also thinks Luke ages T /γ2. But in his frame of reference, his departure is simultaneous with Luke’s event Z, not X, since he has diﬀerent lines of simultaneity. So the asymmetry between Luke and Leia occurs when Leia turns around. At this point, she sees Luke age rapidly from X to Z. Length contraction A rod of length L is stationary in S. What is its length in S? In S, then length of the rod is the distance between the two ends at the same time. So we have In S, we have ct L L x 69 8 Special relativity IA Dynamics and Relativity ct L L x x The lines x = 0 and x = L map into x = vt and x = vt + L/γ. So the length in S is L = L/γ < L. Therefore moving objects are contracted in the direction of motion. Deﬁnition (Proper length). The proper length is the length measured in an object’s rest frame. This is analogous to the fact that if you view a bar from an angle, it looks In relativity, what causes the shorter than if you view it from the front. contraction is not a spatial rotation, but a spacetime hyperbolic rotation. Question: does a train of length 2L ﬁt alongside a platform of length L if it travels through the station at a speed v such that γ = 2? For the system of observers on the platform
, the train contracts to a length 2L/γ = L. So it ﬁts. But for the system of observers on the train, the platform contracts to length L/γ = L/2, which is much too short! This can be explained by the diﬀerence of lines of simultaneity, since length is the distance between front and back at the same time. ctback of train front of train L doesn’t ﬁt in S ﬁts in S back of platform x front of platform Composition of velocities A particle moves with constant velocity u in frame S, which moves with velocity v relative to S. What is its velocity u in S? The world line of the particle in S is In S, using the inverse Lorentz transformation, x = ut. u = x t = γ(x + vt) γ(t + (v/c2)x) = ut + vt t + (v/c2)ut = u + v 1 + uv/c2. 70 8 Special relativity IA Dynamics and Relativity This is the formula for the relativistic composition of velocities. The inverse transformation is found by swapping u and u, and swapping the sign of v, i.e. Note the following: u = u − v 1 − uv/c2. – if uv c2, then the transformation reduces to the standard Galilean addition of velocities u ≈ u + v. – u is a monotonically increasing function of u for any constant v (with \|v\| < c). – When u = ±c, u = u for any v, i.e. the speed of light is constant in all frames of reference. – Hence \|u\| < c iﬀ \|u\| < c. This means that we cannot reach the speed of light by composition of velocities. 8.4 Geometry of spacetime We’ll now look at the geometry of spacetime, and study the properties of vectors in this spacetime. While spacetime has 4 dimensions, and each point can be represented by 4 real numbers, this is not ordinary R4. This can be seen when changing coordinate systems, instead of rotating the axes like in R4, we “squash” the axes towards the diagonal, which is a hyperbolic rotation. In particular, we will have a diﬀerent notion
of a dot product. We say that this space has dimension d = 1 + 3. The invariant interval In regular Euclidean space, given a vector x, all coordinate systems agree on the length \|x\|. In Minkowski space, they agree on something else. Consider events P and Q with coordinates (ct1, x1) and (ct2, x2) separated by ∆t = t2 − t1 and ∆x = x2 − x1. Deﬁnition (Invariant interval). The invariant interval or spacetime interval between P and Q is deﬁned as ∆s2 = c2∆t2 − ∆x2. Note that this quantity ∆s2 can be both positive or negative — so ∆s might be imaginary! Proposition. All inertial observers agree on the value of ∆s2. Proof. c2∆t2 − ∆x2 = c2γ2 = γ2 1 − 2 v c2 ∆x (c2∆t2 − ∆x2) ∆t − v2 c2 − γ2(∆x − v∆t)2 = c2∆t2 − ∆x2. 71 8 Special relativity IA Dynamics and Relativity In three spatial dimensions, ∆s2 = c2∆t2 − ∆x2 − ∆y2 − ∆z2. We take this as the “distance” between the two points. For two inﬁnitesimally separated events, we have Deﬁnition (Line element). The line element is ds2 = c2dt2 − dx2 − dy2 − dz2. Deﬁnition (Timelike, spacelike and lightlike separation). Events with ∆s2 > 0 are timelike separated. It is possible to ﬁnd inertial frames in which the two events occur in the same position, and are purely separated by time. Timelike-separated events lie within each other’s light cones and can inﬂuence one another. Events with ∆s2 < 0 are spacelike separated. It is possible to ﬁnd inertial frame in which the two events occur in the same time, and are purely separated by space. Spacelike-separated events
lie out of each other’s light cones and cannot inﬂuence one another. Events with ∆s2 = 0 are lightlike or null separated. In all inertial frames, the events lie on the boundary of each other’s light cones. e.g. diﬀerent points in the trajectory of a photon are lightlike separated, hence the name. Note that ∆s2 = 0 does not imply that P and Q are the same event. The Lorentz group The coordinates of an event P in frame S can be written as a 4-vector (i.e. 4-component vector) X. We write X =         ct x y z The invariant interval between the origin and P can be written as an inner product X · X = X T ηX = c2t2 − x2 − y2 − z2, where     η = 1 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 −1    . 4-vectors with X · X > 0 are called timelike, and those X · X < 0 are spacelike. If X · X = 0, it is lightlike or null. A Lorentz transformation is a linear transformation of the coordinates from one frame S to another S, represented by a 4 × 4 tensor (“matrix”): X = ΛX Lorentz transformations can be deﬁned as those that leave the inner product invariant: (∀X)(X · X = X · X), 72 8 Special relativity IA Dynamics and Relativity which implies the matrix equation ΛT ηΛ = η. (∗) These also preserve X · Y if X and Y are both 4-vectors. Two classes of solution to this equation are   , where R is a 3 × 3 orthogonal matrix, which rotates (or reﬂects) space and leaves time intact; and  Λ = −γβ 0 0 1 0 c, and γ = 1/1 − β2. Here we leave the y and z coordinates intact, γ −γβ
where β = v and apply a Lorentz boost along the x direction. The set of all matrices satisfying equation (∗) form the Lorentz group O(1, 3). It is generated by rotations and boosts, as deﬁned above, which includes the absurd spatial reﬂections and time reversal. The subgroup with det Λ = +1 is the proper Lorentz group SO(1, 3). The subgroup that preserves spatial orientation and the direction of time is the restricted Lorentz group SO+(1, 3). Note that this is diﬀerent from SO(1, 3), since if you do both spatial reﬂection and time reversal, the determinant of the matrix is still positive. We want to eliminate those as well! Rapidity Focus on the upper left 2 × 2 matrix of Lorentz boosts in the x direction. Write Λ[β] = γ −γβ −γβ γ, γ = 1 1 − β2. Combining two boosts in the x direction, we have Λ[β1]Λ[β2] = γ1 −γ1β1 −γ1β1 γ1 γ2 −γ2β2 −γ2β2 γ2 = Λ β1 + β2 1 + β1β2 after some messy algebra. This is just the velocity composition formula as before. This result does not look nice. This suggests that we might be writing things in the wrong way. We can compare this with spatial rotation. Recall that with R(θ) = cos θ − sin θ sin θ cos θ R(θ1)R(θ2) = R(θ1 + θ2). For Lorentz boosts, we can deﬁne 73 8 Special relativity IA Dynamics and Relativity Deﬁnition (Rapidity). The rapidity of a Lorentz boot is φ such that β = tanh φ, γ = cosh φ, γβ = sinh φ. Then Λ[β] = cosh φ − sinh φ cosh φ − sinh φ = Λ(φ). The rapidities add like rotation angles: Λ(φ1)Λ(φ2) = Λ(φ1 + φ2).
This shows the close relation betweens spatial rotations and Lorentz boosts. Lorentz boots are simply hyperbolic rotations in spacetime! 8.5 Relativistic kinematics In Newtonian mechanics, we describe a particle by its position x(t), with its velocity being u(t) = dx dt. In relativity, this is unsatisfactory. In special relativity, space and time can be mixed together by Lorentz boosts, and we prefer not to single out time from space. For example, when we write the 4-vector X, we put in both the time and space components, and Lorentz transformations are 4 × 4 matrices that act on X. In the deﬁnition of velocity, however, we are diﬀerentiating space with respect to time, which is rather weird. First of all, we need something to replace time. Recall that we deﬁned “proper length” as the length in the item in its rest frame. Similarly, we can deﬁne the proper time. Deﬁnition (Proper time). The proper time τ is deﬁned such that ∆τ = ∆s c τ is the time experienced by the particle, i.e. the time in the particles rest frame. The world line of a particle can be parametrized using the proper time by t(τ ) and x(τ ). ct τ2 τ1 Inﬁnitesimal changes are related by dτ = ds c = 1 c c2 dt2 − \|dx\|2 = 74 x 1 − \|u\|2 c2 dt. 8 Special relativity IA Dynamics and Relativity Thus with dt dτ = γu γu = 1 1 − \|u\|2 c2. The total time experienced by the particle along a segment of its world line is T = dτ = 1 γu dt. We can then deﬁne the position 4-vector and 4-velocity. Deﬁnition (Position 4-vector and 4-velocty). The position 4-vector is X(τ ) = ct(τ ) x(τ ). Its 4-velocity is deﬁned as U = dX dτ = c dt dτ dx dτ = dt dτ c u = γu c
u, where u = dx dt. Another common notation is X = (ct, x), U = γu(c, u). If frames S and S are related by X = ΛX, then the 4-velocity also transforms as U = ΛU. Deﬁnition (4-vector). A 4-vector is a 4-component vectors that transforms in this way under a Lorentz transformation, i.e. X = ΛX. When using suﬃx notation, the indices are written above (superscript) instead of below (subscript). The indices are written with Greek letters which range from 0 to 3. So we have X µ instead of Xi, for µ = 0, 1, 2, 3. If we write Xµ instead, it means a diﬀerent thing. This will be explained more in-depth in the electromagnetism course (and you’ll get more confused!). U is a 4-vector because X is a 4-vector and τ is a Lorentz invariant. Note that dX/dt is not a 4-vector. Note that this deﬁnition of 4-vector is analogous to that of a tensor — things that transform nicely according to our rules. Then τ would be a scalar, i.e. rank-0 tensor, while t is just a number, not a scalar. For any 4-vector U, the inner product U · U = U · U is Lorentz invariant, i.e. the same in all inertial frames. In the rest frame of the particle, U = (c, 0). So U · U = c2. In any other frame, Y = γu(c, u). So Y · Y = γ2 u(c2 − \|u\|2) = c2 as expected. 75 8 Special relativity IA Dynamics and Relativity Transformation of velocities revisited We have seen that velocities cannot be simply added in relativity. However, the 4-velocity does transform linearly, according to the Lorentz transform: U = ΛU. In frame S, consider a particle moving with speed u at an angle θ to the x axis in the xy plane. This is the most general case for motion not parallel to the Lorentz boost. Its 4-velocity is U =  �
��   γuc γuu cos θ γuu sin θ 0    , γu = 1 1 − u2/c2. With frames S and S in standard conﬁguration (i.e. origin coincide at t = 0, S moving in x direction with velocity v relative to S), U =     γuc γuu cos θ γuu sin θ 0     =     γv −γvv/c 0 0 −γvv/c 0 0 1 0 γv γuc γuu cos θ γuu sin θ 0     Instead of evaluating the whole matrix, we can divide diﬀerent rows to get useful results. The ratio of the ﬁrst two lines gives u cos θ = u cos θ − v 1 − uv c2 cos θ, just like the composition of parallel velocities. The ratio of the third to second line gives tan θ = u sin θ γv(u cos θ − v), which describes aberration, a change in the direction of motion of a particle due to the motion of the observer. Note that this isn’t just a relativistic eﬀect! If you walk in the rain, you have to hold your umbrella obliquely since the rain seems to you that they are coming from an angle. The relativistic part is the γv factor in the denominator. This is also seen in the aberration of starlight (u = c) due to the Earth’s orbital motion. This causes small annual changes in the apparent positions of stars. 4-momentum Deﬁnition (4-momentum). The 4-momentum of a particle of mass m is P = mU = mγu c u 76 8 Special relativity IA Dynamics and Relativity The 4-momentum of a system of particles is the sum of the 4-momentum of the particles, and is conserved in the absence of external forces. The spatial components of P are the relativistic 3-momentum
, p = mγuu, which diﬀers from the Newtonian expression by a factor of γu. Note that \|p\| → ∞ as \|u\| → c. What is the interpretation of the time component P 0 (i.e. the ﬁrst time component of the P vector)? We expand for \|u\| c: P 0 = mγc = mc 1 − \|u\|2/c2 = 1 c mc2 + 1 2 m\|u\|2 + · · ·. We have a constant term mc2 plus a kinetic energy term 1 2 m\|u\|2, plus more tiny terms, all divided by c. So this suggests that P 0 is indeed the energy for a particle, and the remaining · · · terms are relativistic corrections for our old formula 1 2 m\|u\|2 (the mc2 term will be explained later). So we interpret P as P = E/c p Deﬁnition (Relativistic energy). The relativistic energy of a particle is E = P 0c. So E = mγc2 = mc2 + Note that E → ∞ as \|u\| → c. For a stationary particle, we obtain 1 2 m\|u\|2 + · · · E = mc2. This implies that mass is a form of energy. m is sometimes called the rest mass. The energy of a moving particle, mγuc2, is the sum of the rest energy mc2 and kinetic energy m(γu − 1)c2. Since P · P = E2 c2 − \|p\|2 is a Lorentz invariant (lengths of 4-vectors are always Lorentz invariant) and equals m2c2 in the particle’s rest frame, we have the general relation between energy and momentum E2 = \|p\|2c2 + m2c4 In Newtonian physics, mass and energy are separately conserved. In relativity, mass is not conserved. Instead, it is just another form of energy, and the total energy, including mass energy, is conserved. Mass can be converged into kinetic energy and vice versa (e.g. atomic bombs!) Massless particles Particles with zero mass (m = 0), e.g. photons, can have non-zero momentum and energy because they travel at the speed of light (γ = ∞). In this case,
P · P = 0. So massless particles have light-like (or null) trajectories, and no proper time can be deﬁned for such particles. 77 8 Special relativity IA Dynamics and Relativity Other massless particles in the Standard Model of particle physics include the gluon. For these particles, energy and momentum are related by So Thus E2 = \|p\|2c2. E = \|p\|c. P = 1 n, E c where n is a unit (3-)vector in the direction of propagation. According to quantum mechanics, fundamental “particles” aren’t really particles but have both particle-like and wave-like properties (if that sounds confusing, yes it is!). Hence we can assign it a de Broglie wavelength, according to the de Broglie relation: \|p\| = h λ where h ≈ 6.63 × 10−34 m2 kg s−1 is Planck’s constant. For massless particles, this is consistent with Planck’s relation: E = hc λ = hν, where ν = c λ is the wave frequency. Newton’s second law in special relativity Deﬁnition (4-force). The 4-force is F = dP dτ This equation is the relativistic counterpart to Newton’s second law. It is related to the 3-force F by F = γu F · u/c F Expanding the deﬁnition of the 4-force componentwise, we obtain and dE dτ = γuF · u ⇒ dp dτ = γuF ⇒ dE dt dp dt = F · u = F Equivalently, for a particle of mass m, where F = mA, A = dU dτ 78 8 Special relativity IA Dynamics and Relativity is the 4-acceleration. We have So U = γu c u A = γu dU dt = γu ˙γuc γua + ˙γuu. where a = du dt and ˙γu = γ3 u a·u c2. In the instantaneous rest frame of a particle, u = 0 and γu = 1. So U = c 0, A = 0 a Then U · A = 0. Since this is a
Lorentz invariant, we have U · A = 0 in all frames. 8.6 Particle physics Many problems can be solved using the conservation of 4-momentum, P = E/c p, for a system of particles. Deﬁnition (Center of momentum frame). The center of momentum (CM) frame, or zero momentum frame, is an inertial frame in which the total 3-momentum is p = 0. This exists unless the system consists of one or more massless particle moving in a single direction. Particle decay A particle of mass m1 decays into two particles of masses m2 and m2. We have i.e. P1 = P2 + P3. E1 = E2 + E3 p1 = p2 + p3. In the CM frame (i.e. the rest frame of the original particle), E1 = m1c2 = \|p2\|2c2 + m2 ≥ m2c2 + m3c2. 2c4 + \|p3\|2c2 + m2 2c4 So decay is possible only if m1 ≥ m2 + m3. (Recall that mass is not conserved in relativity!) 79 8 Special relativity IA Dynamics and Relativity Example. A possible decay path of the Higgs’ particle can be written as h → γγ Higgs’ particle → 2 photons This is possible by the above criterion, because mh ≥ 0, while mγ = 0. The full conservation equation is So Ph = mhc 0 = Pγ1 + Pγ2 pγ1 = pγ2 Eγ1 = Eγ2 = 1 2 mhc2. Particle scattering When two particles collide and retain heir identities, the total 4-momentum is conserved: P1 + P2 = P3 + P4 In the laboratory frame S, suppose that particle 1 travels with speed u and collides with particle 2 (at rest). 1 2 θ φ 1 2 In the CM frame S, 2 = 0 = p Both before and after the collision, the two particles have equal and opposite 3-momentum. 1 + p p 3 + p 4. p3 1 p2 2 1 p1 p4 2 The scattering angle θ is undetermined and can be thought of as being random. However, we can derive some conclusions about the angles θ and φ in
the laboratory frame. 80 8 Special relativity IA Dynamics and Relativity (staying in S for the moment) Suppose the particles have equal mass m. They then have the same speed v in S. Choose axes such that     mγvc mγvv 0 0    , P 2 =     mγvc −mγvv 0 0     P 1 = and after the collision, P 3 =     mγvc mγvv cos θ mγvv sin θ 0    , P 4 =     mγvc −mγvv cos θ −mγvv sin θ 0    . We then use the Lorentz transformation to return to the laboratory frame S. The relative velocity of the frames is v. So the Lorentz transform is Λ =     γv γvv/c 0 0 γvv/c 0 0 1 0 γv and we ﬁnd where     P1 =     mγuc mγuu 0 0, P2 = u = 2v 1 + v2/c2,         mc 0 0 0 (cf. velocity composition formula) Considering the transformations of P 3 and P 4, we obtain tan θ = sin θ γv(1 + cos θ) = 1 γv tan and 1 γv Multiplying these expressions together, we obtain sin θ γv(1 − cos θ) tan φ = = cot θ 2, θ 2. tan θ tan φ = 1 γ2 v. So even though we do not know what θ and φ might be, they must be related by this equation. In the Newtonian limit, where \|v\| c, we have γv ≈ 1. So i.
e. the outgoing trajectories are perpendicular in S. tan θ tan φ = 1, 81 8 Special relativity IA Dynamics and Relativity Particle creation Collide two particles of mass m fast enough, and you create an extra particle of mass M. P1 + P2 = P3 + P4 + P5, where P5 is the momentum of the new particle. In the CM frame, v 1 v 2 P1 + P2 = 2mγvc 0 P3 + P4 + P5 = (E3 + E4 + E5)/c 0 We have So 2mγvc2 = E3 + E4 + E5 ≥ 2mc2 + M c2. So in order to create this new particle, we must have γv ≥ 1 + M 2m. Alternatively, it occurs only if the initial kinetic energy in the CM frame satisﬁes 2(γv − 1)mc2 ≥ M c2. If we transform to a frame in which the initial speeds are u and 0 (i.e. stationary target), then u = 2v 1 + v2/c2 Then So we require γu = 2γ2 v − 1. γu ≥ 2 1 + 2 M 2m − 1 = 1 + 2M m + M 2 2m. This means that the initial kinetic energy in this frame must be m(γu − 1)c2 ≥ 2 + M 2m M c2, which could be much larger than M c2, especially if M m, which usually the case. For example, the mass of the Higgs’ boson is 130 times the mass of the proton. So it would be much advantageous to collide two beams of protons head on, as opposed to hitting a ﬁxed target. 82the following arc true for all sets A, B, and C? (a) IfA V Band B(ZC, thenA IZC. (b) If A B and B C, then A Pd- C. (c) If A C B and / C, then A 1Z C. (d) If A C Band BCC, then C CT- A. (e) If A C Band B C C, then A VC. 3.5. Show that for every set A, A C 0 if A = 0. 3.6. Let A,, A2,., A. be n sets
. Show that A,CA2C... if Ar=A2=... =A,,. 3.7. Give several examples of a set X such that each element of X is a subset of X. 3.8. List the members of 61(A) if A = {{1, 2), {3), 1). 3.9. For each positive integer n, give an example of a set A. of n elements such that for each pair of elements of A,,, one member is an element of the other. 4. Operations for Sets We continue with our description of methods for generating new sets from existing sets by defining two methods for composing pairs of sets. These so-called operations for sets parallel, in certain respects, the familiar operations of addition and multiplication for integers. The union (sum, join) of the sets A and B, symbolized by A U 13 and read "A union B" or "A cup B," is the set of all objects which are members of either A or B; that is, A U B = {x1xrC A or x C B). Here the inclusive sense of the word "or" is intended. Thus, by defi- 1.4 I Operations for Sets 13 nition, x E A U B ifr x is a member of at least one of A and B. For example, 11, U {1,3,4) = 11,2,3,41. The intersection (product, meet) of the sets A and 13, symbolized by A n B and read "A intersection 13" or "A cap B," is the set of all objects which are members of both A and B; that is, AnB= {x{xCAandxCB). Thus, by definition, x C An B iff x C A and x C B. For example, {1,2,31 n {l,3,4) = {1,31. It is left as an exercise to prove that for every pair of sets A and B the following inclusions hold : 09;AnB_AcAUB. Two sets A and B are disjoint iff A n 13 = 0, and they intersect iff A n B 0. A collection of sets is a disjoint collection iff each distinct pair of its member sets is disjoint. A partition of a set X is a disjoint collection a of noncrnpty and distinct subsets of
X such that each member of X is a member of some (and, hence, exactly one) member of a. For example, { 11, 21, 131, 14, 5) 1 is a partition of 11,2,3,4,5). A further procedure, that of complementation, for generating sets from existing sets employs a single set. The absolute complement of a set A, symbolized by A, is {xix (Z A). The relative complement of A with respect to a set X is X n A; this is usually shortened to X - A, read "X minus A." 'Thus X - A = {xCXJx(Z A), that is, the set of those members of X which are not members of A. The symmetric difference of sets A and B, symbolized by A ± B, is defined as follows : A±B=(A-B)U(B-A). This operation is commutative, that is, A + B = B + A, and associative, that is, (A + B) + C = A + (B + C). Further, A + A = 0, and A + 0 = A. Proofs of these statements are left as exercises. If all sets under consideration in a certain discussion arc subsets of a set U, then U is called the universal set (for that discussion). As examples, in elementary number theory the universal set is Z, and in 14 Sels and Relations I C H A P. I plane analytic geometry the universal set is the set of all ordered pairs of real numbers. A graphic device known as a Venn diagram is used for assisting one's thinking on complex relations which may exist among subsets of a universal set 11. A Venn diagram is a schematic representation of scats by sets of points: the universal set U is represented by the points within a rectangle, and a subset A of U is represented by the interior of a circle or some other simple region within the rectangle. The complement of A relative to If, which we may abbreviate to A without confusion, is the part of the rectangle outside the region representing A, as shown in Figure 1. If the subsets A and 13 of U are repre- A shaded Figure 1 Any shaded Figure 2 AUB shaded Figure 3 sented in this way, then A n B and A U B are represented by shaded regions, as in Figure 2 and Figure 3, respectively.
Disjoint sets are represented by nonoverlapping regions, and inclusion is depicted by displaying one region lying entirely within another. These are the ingredients for constructing the Venn diagram of an expression compounded from several sets by means of union, intersection, complementation, and inclusion. The principal applications of Venn diagrams are to problems of simplifying a given complex expression and simplifying given sets of conditions among several subsets of a universe of discourse. Three simple examples of this sort appear below. In many cases such diagrams are inadequate, but they may be helpful in connection with the algebraic approach developed in the next section. EXAMPLES 4.1. Suppose A and B are given sets such that A - B = B - A = 0. Can the relation of A to B be expressed more simply? Since A - B = 0 means 0, the regions representing A and T? do not overlap (Figure 4). A f T3 Clearly, B = B, so we conclude (Figure 5) that A _C B. Conversely, if A C B, it is clear that A - B = 0. We conclude that A - B = 0 iff A C B. Interchanging A and B gives B - A = 0 iff B C A. Thus the given relations hold betweenAandBif-fACBandBCAor,A=B. 1.4 I Operations for Sets 15 Figure 4 Figure 5 Figure 6 4.2. Let us investigate the question of whether it is possible to find three sub- sets A, B, and C of U such that C00,AnB5- 0,AnC=o,(AfB)-C=Qf. The second condition implies that A and B intersect and, therefore, incidentally that neither is empty. From Example 4.1 the fourth condition amounts to A fl B C C, from which it follows that the first is superfluous. The associated Venn diagram indicates that A and C intersect; that is, the validity of the second and fourth conditions contradicts the third. Hence, there do not exist sets satisfying all the conditions simultaneously. 4.3. Given that F, C, and L are subsets of U such that F9 G,G(1 LCF,Lf1F=0. Is it possible to simplify this set of conditions? The Venn diagram (Figure 6) represents only the first and third conditions. The second condition forces L and C to be disjoint, that is,
G fl L = 0. On the other hand, if F C C and G fl L = 0, then all given conditions hold. Thus F C G and G fl L = 0 constitute a simplification of the given conditions. EXERCISES (Note: Venn diagrams are not to be used in Exercises 4.1-4.8.) 4.1. I'rove that for all sets A and B, 0 A f B9. A U B. 4.2. Let I be the universal set, and let A = {x C LI for some positive integer y, x = 2y}, B = {x C 1I for some positive integer y, x = 2y - 1), C = {xCZIx < 10). Describe A, A _U B, c, A - C, and C - (A U B), either in prose or by a defining property. 4.3. Consider the following subsets of Z1, the set of positive integers: A = {x C "G_}i for some integer y, x = 2y}, B = {x C 7--1 1 for some integer y, x = 2y + 1), C = {x C `L_i-I for some integer y, x = 3y}. (a) DescribeAflC, B U C, and B - C. (b) Verify that A fl (B U C) = (A (1 B) U (An C). 16 Sets and Relations I CHAP P. 1 4.4. If A is any set, what are each of the following sets? A n o, A U 0, 4.5. Determine o n (0), {O) n {O), {O, {O}} - 0, {0, (0}} - A - 0, A - A, 0 - A. (0), {0, (0)) - ((0)). 4.6. Suppose A and B are subsets of U. Show that in each of (a), (b), and (c) below, if any one of the relations stated holds, then each of the others holds. (a) ACB,21 QIi,AU13= B, AnB=A. (b) AnB= 0,AC F3, BCI1. (c) AUB= U,ACB,f3CA. 4.
7. Prove that for all sets A, B, and C, (AnB) UC= An (BU C) if CCA. 4.8. Prove that for all sets A, B, and C, (A - 13) - C= (A - C) - (B-C). 4.9. (a) Draw the Vcnn diagram of the symmetric difference, A + B, of sets A and B. (b) Using a Venn diagram, show that symmetric difference is a com- mutative and associative operation. (c) Show that for every set A, A + A = 0 and A + 0 = A. 4.10. The Venn diagram for subsets A, B, and C of U, in general, divides the rectangle representing U into eight nonoverlapping regions. Label each region with a combination of A, B, and C which represents exactly that region. 4.11. With the aid of a Venn diagram investigate the validity of each of the following inferences: (a) If A, B, and C are subsets of U such that A n B C C and A U C C B, (b) if A, B, and C are subsets of U such that A C B -U C and B C A 3 C, then AfC=0. then B = 0. 5. The Algebra of Sets If we were to undertake the treatment of problems more complex than those examined above, we would feel the need for more systematic procedures for carrying out calculations with sets related by inclusion, union, intersection, and corriplementation. That is, what would be called for could appropriately be named "the algebra of sets"-a development of the basic properties of U, n,, and C together with interrelations. As such, the algebra of sets is intended to be the settheoretic analogue of the familiar algebra of the real numbers, which is concerned with properties of +,, and < and their interrelations 1.5 1 The Algebra of Sets 17 The basic ingredients of the algebra of sets are various identitiesequations which are true whatever the universal set U and no matter what particular subsets the letters (other than U and 0) represent. Our first result lists basic properties of union and intersection. For the sake of uniformity, all of these have been formulated for subsets of a universal set U. However, for some of the properties this is a
purely artificial restriction, as an examination of the proofs will show. THEOREM 5.1. For any subsets A, B, C of a set U the following equations are identities. Here t1 is an abbreviation for U - A. 1.AU(BUC) 2. AUB=BUA. 3.AU(Bnc) =(AUB)UC. = (A U B) n(A U C). 1'.An(Bnc) 2'. AnB=BnA. 3'.An(BUC) =(AnB)nC. =(AnB)U(Anc). 4. AU 0 A. 5.AUt1=U. Proof. Each assertion can be verified by showing that the set on either side of the equality sign is include([ in the set on the other side. As an illustration we shall prove identity 3. 4'. An U=A. 5'.An:=0. (a) Proof that A U (B n C) S (A U B) n (A U C). Let x C A U (BnC).Then xEAor xEBnC.If xCA,then xCAUB and x C A U C, and hence x is a member of their intersection. If xCBnC., then xC BandxEC.IlencexCA U B and x E A U C, so again x is a member of their intersection. (b) Proof that (A U B) n (A U C) C A U (BnC). Let x C (AUB)n(AUC).Then xEAUBand xEAUC.Hence x C A, or x E B and x C C. These imply that x C A U (B n C). Identities 1 and 1' are referred to as the associative laws for union and intersection, respectively, and identities 2 and 2' as the commutative laws for these operations. Identities 3 and 3' are the distributive laws for union and intersection, respectively. The analogy of properties of union and intersection with properties of addition and znultiplication, respectively, for numbers, is striking at this point. For instance, 3' corresponds precisely to the distributive law in arithmetic. That there are also striking differences is illustrated by 3, which has no analogue in arithmetic. 18 Sets and Relations I G H A P. 1 According to the associative law
, identity 1, the two sets that can be formed with the operation of union from sets A, B, and C, in that order, are equal. We agree to denote this set by A U B U C. Then the associative law asserts that it is immaterial as to how parentheses are introduced into this expression. Using induction, this result can be generalized to the following. The sets obtainable frorn given sets A,,, An, in that order, by use of the operation of union are all equal A2, to one another. The set defined by A,, A2,, A. in this way will be written as A,UA2U UA.. In view of identity I' there is also a corresponding generalization for intersection. With these general associative laws on the record we can state the general commutative law: If 1', 2', n in any order, then, n' are 1, 2, A,UA2U... UA4=A1.UA2.U... UA,,,. We can also state the general distributive laws: 112fl... r) B.) Afl(B,UB2U... = (A U fl (A U B2) n... fl (A U U (A U (A fl B.). 'T'hese can also be proved by induction. Detailed proofs of the foregoing properties of unions and intersections of sets need make no reference to the membership relation; that is, these properties follow solely from those listed in Theorem 5.1. The same is true of those further properties which appear in the next theorem. Such facts may be regarded as the origin of the "axiomatic approach" to the algebra of sets developed in Chapter 6. One derivative of this approach is the conclusion that every theorem of the algebra of sets is derivable from 1 5 and 1'-5'. These ten properties have another interesting consequence. In Theorem 5.1 they are paired in such a way that each member of a pair is obtainable from the other member by interchanging U and fl and, simultaneously, 0 and U. An equation, or an expression, or a statement within the framework of the algebra of sets obtained from another by interchanging U and fl along with 0 and U throughout is the dual of the original. We contend that the dual of any theorem expressible in terms of U, fl, and, and which can be proved using only identities 1-5 and 1'--5', is also a theorem
. Indeed, suppose that the proof of such a theorem is written as a sequence of steps and that 1.5 1 The Algebra of Sees 19 opposite each step is placed the justification for it. By assumption, each justification is one of 1-5, one of 1'- 5', or a premise of the theorem. Now replace the identity or relation in each step by its dual. Since 1-5 and 1'--5' contain with each its dual, and the dual of each premise of the original theorem is now a premise, the dual of each justification in the original proof is available to serve as a justification for a step in the new sequence which, therefore, constitutes a proof. The last line of the new sequence is, therefore, a theorem, the dual of the original theorem. Accepting the fact that every theorem of the algebra of sets is deducible from 1---5 and 1'--5', we then obtain the principle of duality for the algebra of sets: If 7' is any theorem expressed in terms of U, 1,, then the dual of T is also a theorem. This implies, for instance, and that if the unprimed formulas in the next theorem are deduced solely from Theorem 5.1, then the primed formulas follow by duality. The reader should convince himself that all the assertions in 't'heorem 5.2 are true by using the definitions of U, fl, and in terms of the membership relation. Further, lie might try to deduce some of them solely from Theorem 5.1-that is, without appealing in any way to the membership relation. Some demonstrations of this nature appear in the proof of Theorem 6.2.1. t TIIEOREM 5.2. For all subsets A and B of a set U, the following statements are valid. here A is an abbreviation for U - A. 6. If, for all A, A U B = A, 6'. If, for all A,Al II=A,then then B = 0. B = If. 7, 7'. If A U B = U and A flB= 0, then B 8, 8'. A=A. = U. 9. 10. AUA=A. 11. A U U U. 12. A U (A n B) = A. 13. AU11=Afl). 9'. I> _ 0. 10'. Af1A=A. 11'.
A fl 0 12'. 'A fl (A U B) = A. 13'. Af1B=AU13. 0. Some of the identities in Theorem 5.2 have well-established names. For example, 10 and 10' are the idempotent laws, 12 and 12' are the t To refer to a theorem, example, exercise, or section in the chapter in which it appears, we use only the number by which it is identified in the text. When a reference is made to one of these items in another chapter we prefix its identifying number with a numeral that identifies the chapter. For instance, in Chapter t we shall refer to the third example in Section 2 as Example 2.3 and in another chapter we shall refer to the same example as Lxample 1.2.3. 20 Sets and Relations I CA IAP. 1 absorption laws, and 13 and 13' the DeMorgan laws. The identities 7, 7' and 8, 8' are each numbered twice to emphasize that each is unchanged by the operation which converts it into its dual; such formulas arc called self-dual. Note that 7, 7' asserts that each set has a unique complement. A remark about the form of time next theorem is in order. An assertion of the form, "The statements R,, B2,.., RA, arc equivalent to one another," means "For all i and j, R; if R;," which, in turn, is the case, Rk-t implies Rk, and Rk implies Rt. The iff R, implies Rz implies R3, content of the theorem is that the inclusion relation for sets is definable in terms of union as well as in terms of intersection. THEOREM 5.3. The following statements about sets A and B are equivalent to one another. (1) Ac 11. (II)AfB=A. (11I) AU13=B. (I) implies (11). Assume that A C B. Since, for all A and B, Proof. A nB c A, it is sufficient to prove that A C A fl I3. But if x C A, then x C B and, hence, x E A fl B. Hence A C A fl B. (II) implies (III). Assume A fl B = A. Then AUB=(AfIB)UB= (AUB)fl(I3UB) _ (AU
1.1) f113=B. (1II) implies (1). Assume that A U B = B. Then this and the iden- tity AcAUBimply AcB. The principle of duality as formulated earlier does not apply directly to expressions in which - or C_ appears. One can cope with subtraclion by using the unabbreviated form, namely, A fl 73, for A - B. Similarly, by virtue of 'Theorem 5.3, A C 13 may be replaced by A fl B = A (or A U B = 13). Still better, since the dual of A fl 13 = A is A U B = A, which is equivalent to A -D 13, the principle of duality may be extended to include the case where the inclusion symbol is present, by adding the provision that all inclusion signs be reversed. EXAMPLES 5.1. With the aid of the identities now available a great variety of complex expressions involving sets can be simplified, much as in elementary algebra. We give three illustrations. 1.5 f The Algebra of,Sets 21 (a) AfBUB=AU BUB (b) (An L3nC) U (A () B(l C) U IJUC U B. =I(AUA)nB()c]uRUC =[Un13nC] UBnC (BnC)UIIT C U. (c) (AnBnCn)U(Anc)U(inc)U(CnX) (An BnCnx)U [(AU 13UX) nC] = Unc = C. 5.2. There is a theory of equations for the algebra of sets, and it differs considerably from that encountered in high school algebra. As an illustration we shall discuss a method for solving a single equation in one "unknown." Such an equation may be described as one formed using (l, U, and on symbols., A,,, and A', where the A's denote (fixed subsets of some universal set A,) A2, 11 and X denotes a subset of U which is constrained only by the equation in which it appears. Using the algebra of sets, the problem is to determine under what conditions such an equation has a solution and then, assuming these arc satisfied, to obtain all solutions. A recipe for this follows; the proof required in each step is left as an
exercise (see Exercise 5.7). Step 1. Two sets are equal iff their symmetric difference is equal to 0. Hence, all equation in X is equivalent to one whose righthand side is 0. Step If. An equation in X with righthand side 0 is equivalent to one of the form (AfX)U(Bn1) =0, where A and 13 are free of X. Step Ill. The union of two sets is equal to Qf iff each set is equal to 0. I lence, the equation in Step II is equivalent to the pair of simultaneous equations AnX=0, Bnx=0. Step IV. The above pair of equations, and hence the original equation, has a solution iff 13 C_ A. In this event, any A', such that B C X C A, is it soluctioti. We illustrate the foregoing by deriving necessary and sufficient conditions that the following equation have a solution: X U C = 1), [(.1 UC)nT]U[vn(,1UG)j-0, [(XUC)nl)]U[Dn xncj= 0, (,1'n_ID)U(cnT))U(D nTnC)=0, (Dnx)U[(CnD)n(XUx)]U(UnCnx) =0. (Step I) 22 Sets and Relations I C II A P. 1 (The introduction of X U X in the preceding equation is discussed in Exercise 5.7.) (13nX)U(CnTnX)U(cnDnT)U(DnEn T) = 0, {[DU(Cnl))]nX}U{[(Cn75)U(DfC)Inxi =0, (D n X) U [(C + 1)) n xi = 0, (Step II) 7)fX=0and (C+D)fX=0. (Step 111) Thus, the original equation has a solution ill C + 1) C D. (Step IV) It is left as an exercise to show that this condition simplifies to C G D. EXERCISES 5.1. Prove that parts 3', 4', and 5' of Theorem 5.1 are identities. 5.2. Prove the unprimed parts of
Theorem 5.2 using the membership relation. Try to prove the same results using only Theorem 5.1. In at least one such proof write out the dual of each step to demonstrate that a proof of the dual results. following equations is an identity. 5.3. Using only the identities in Theorems 5.1 and 5.2, show that each of the (a) (AnBfX)U(Af BnCfXfY)U(AnXnA) =AfBnX. (b) (A n RnC)U (1nBnC)UPUC= U. (c) (AntinCn )u(AnC)u(7nc)u(CnX)=C. (d) [(Af B)U(AnC)U(AnXnY)] n[(Ar) 1nc)U(Ai1X(1Y)u(AnBnY)] (AfB)Uc! nli'nXnY). 5.4. Rework Exercise 4.9(b), using solely the algebra of sets developed in this section. 5.5. Let A,, A2,, A. be sets, and define Sk to be A, U A2 U U At for A = 1, 2,, n. Show that (3,= is a disjoint collection of sets and that When is (i a partition of S,? 5.6. Prove that for arbitrary sets A,, A2, U(A.-S,._,)., 2), A,UA2U...UA,.=(A,-A2)U(A2-A3)U...U(A.A.) U (A - A,) U (A, n A2 n... n AA). 5.7. Referring to Example 5.2, prove the following. (a) For all sets A and B, A = B ifi' A + B = 0. (b) An equation in X with righthand member 0 can be reduced to one of- 1.6 I Relalions 23 the form (An x) U (B n X) = 0. (Suggestion: Sketch a proof along these lines. First, apply the DeMorgan laws until only complements of* individual sets appear. Then expand the resulting lefthancl side by the distributive law 3 so
as to transform it into the union of several terms T,, each of which is an intersection of several individual sets. Next, if in any T, neither X nor Y appears, replace T, by 7, n (X U X) and expand. Finally, group together the terms containing X and those containing 7i' and apply the distributive law 3'.) (c) For all sets AandB,A=B=0iff'AUB=0. (d) The equation (A n x) U (I3 n Y) = 0 has a solution if 11 C A1, and then any X such that B C X C A is a solution. (e) An alternative form for solutions of the equation in part (d) is X = (B U T) fl A, where 1' is an arbitrary set. 5.8. Show that for arbitrary sets A, B, C, 1), and A, (a) [(A n X) U n U n (c) [(An X)U(I3nA)Jn[(Cn AA)U(nnX)i X)J = [(AUC)nxiu[(BUD)nXJ, I(Af-C)fX]UI(BnD)nXJ. 5.9. Using the results in Exercises 5.7 and 5.8, prove that the equation (AnX)u(ItnX)= (CnX)u(I)nA) has a solution iff B -1- I) C A -}- C. In this event determine all solutions. 6. Relations In mathematics the word "relation" is used in the sense of relationship. The following partial sentences (or predicates) are examples of relations: is less than, divides, is congruent to, is included in, is a member of, is the mother of. In this section the concept of a relation will be developed within the framework of set theory. The motivation for the forthcoming definition is this: A (binary) relation is used in connection with pairs of objects considered in a definite order. Further, a relation is concerned with the existence or nonexistence of sonic type of bond between certain ordered pairs. We infer that a relation provides a criterion for distinguishing some ordered pairs from others in the following sense. 11'a list of all ordered pairs for which the relation is pertinent is available, 24 Sels and Relations
I C II A P. I then with each may be associated "yes" or "no" to indicate that a pair is or is not in the given relation. Clearly, the sane end is achieved by listing exactly all those pairs which are in the given relation. Such a list characterizes the relation. Thus the stage is set for defining a relation as a set of ordered pairs, and this is done as soon as the notion of an ordered pair is made precise. Intuitively, an ordered pair is simply an entity consisting of two objects in a specified order. As the notion is used in mathematics, one relies on ordered pairs to have two properties: (i) given any two objects, x and y, there exists an object, which might be denoted by (x, y) and called the ordered pair of x and y, that is uniquely determined by x and y; (ii) if (x, y) and (u, v) are two ordered pairs, then (x, y) = (u, v) itT x = it and y = v. Now it is possible to define an object, indeed, a set, which has these properties: the ordered pair of x and y, symbolized by is the set (x) y), I {x}, {x, y} }, that is, the two-clement set one of whose members, {x, y), is the unordered pair involved, and the other, {x}, determines which member of this unordered pair is to be considered as being "first." We shall now prove that, as defined, ordered pairs have the properties mentioned above. THEOREM 6.1. The ordered pair of x and y is uniquely detcrrnined by x and y. Moreover, if (x, y) = (u, v), then x = u and y = v. Proof. That x and y uniquely determine (x, y) follows from our assumption that a set is uniquely determined by its members. Turning to the more profound part of the proof, let us assume that (x, y) (u, v). We consider two cases. (1) it = v. Then (u, v) = { {u}, {,,, v} { Jul }. Ihcn('e { {.x}, {x,y} } = { {u} }, which implies that {x} _ {x, y) _ {u} and, in turn,
that x=it and y=v. (II) n p`- v. Then {u} 5,4- {u, v} and {x} {u, v}. Since {x} C { {u}, {u, v) }, it follows that {x} = Jul arid, hence, x = u. Since {u,,)} C fix), {x, y} } and {u, v} - {x}, we have {u, v} = {x, y}. Thus, {x} 76 (x, y), so, in turn, x F y and y 0 it. I y = v. We call x the first coordinate and y the second coordinate of the ordered pair (x, y). Ordered triples and, in general, ordered n-tuplcs 1.6 I Relations 25 may be defined in terms of ordered pairs. The ordered triple of x, y, and z, symbolized by (x, y, z), is defined to be the ordered pair ((x, y), z). Assuming that ordered (n - 1)-tuplcs have been defined, we take the ordered n-tuple of x,, x2,, xn), to be ((x,, x2,..., xn-1), xn), x,,, symbolized by (x,, x2, We return to our principal topic by defining a binary relation as a set of ordered pairs, that is, a set each of whose members is an ordered pair. If p is a relation, we write (x, y) C p and xpy interchangeably, and we say that x is p-related to y iff xpy. There are established symbols for various relations such as equality, membership, inclusion, congruence. Such familiar notation as x = y, x < y, and x = y is the origin of xpy as a substitute for "(x, y) C p." A natural generalization of a binary relation is that of an n-ary relation as a set of ordered n-tuplcs. The case n = 2 is, of course, the one for which we have agreed on the name "binary relation." Similarly, in place of 3-ary relation we shall say ternary relation. EXAMPLES 6.1. {(2, 4), (7, 3), (3, 3), (2, 1)) as a set of ordered
pairs is a binary relation. '1'hc fact that it appears to have no particular significance suggests that it is not worthwhile assigning a name to. 6.2. The relation "less than" for integers is {(x, y)I for integers x and y, there is a positive integer z for which x + z = y}. Symbolizing this relation in the traditional way, the statements "2 < 5" and "(2, 5) C <" are synonymous (and true). 6.3. If i symbolizes the relation of motherhood, then (,Jane,.John) C µ means that.Jane is the mother of John. 6.4. Human parenthood is an example of a ternary relation. If it is symbolized by p, then (Elizabeth, Philip, Charles) C p indicates that Elizabeth and I'hilip are the parents of Charles. Addition in Z is another ternary relation; writing "5 == 2 -I- 3" may be considered as an alternative to asserting that (2, 3, 5) C -f- 6.5. The cube root relation for real numbers is ((x113,x)I x C R). One mcm- ber of this relation is (2, 8). 6.6. In trigonometry the sine function is defined by way of a rule for associating with each real number a real number between - I and 1. In practical applications one relics on a table in a handbook for values of this function for various arguments. Such a table is simply a compact way of displaying a set of ordered pairs. Thus, for practical purposes, the sine function is defined by the set of ordered pairs exhibited in a table (together with a rule concerning the extension of the table). We note that as such a table is designed to be read it 26 Sets and Relations I CHAP. 1 presents pairs of the form (x, sin x); thereby the coordinates are interchanged from the order in which we have been writing them for relations in general. That is, for an arbitrary relation p we have interpreted (a, b) C p as meaning that a is p-related to b, whereas the presence of (7r/2, 1) in a table for the sine function is intended to convey the information that the second coordinate is sine-related (is the sine of) the first coordinate. Later we shall find extensive applications for ternary relations, but
our present interest is in binary relations, which we shall abbreviate to simply "relations" if no confusion can result. If p is a relation, then the domain of p, symbolized by D, is and the range of p, symbolized by R, is {xl for some y, (x, y) E p {yj for some x, (x, y) E p1. That is, the domain of p is the set whose members are the first coordinates of members of p, and the range of p is the set whose members are the second coordinates of members of p. For example, the domain and range of the inclusion relation for subsets of a set If are each equal to a'(U). Again, the domain of the relation of motherhood is the set of all mothers, and the range is the set of all people. One of the simplest types of relations is the set of all pairs (x, y), such that x is a member of some fixed set X and y is a member of some fixed set Y. This relation is the cartesian product, X X Y, of X and Y. Thus, XX Y= {(x,y)jxCXandyC Y}. It is evident that a relation p is a subset of any Cartesian product X X Y, such that X? 1), and Y Q R. If p is a relation and p C X X Y, then p is referred to as a relation from X to Y. If p is a relation from X to Y arrd2 _> X U Y, then p is a relation from 2 to L. A relation from L tot, will be called a relation in L. Such terminologies as "a relation from X to Y" and "a relation in Z" stem from the possible application of a relation to distinguish certain ordered pairs of objects from others. If X is a set, then X X X is a relation in X which we shall call the universal relation in X; this is a suggestive name, since, for each pair x, y of elements in X, we have x(X X X)y. At the other extreme is the void relation in X, consisting of the empty set. Intermediate is the identity relation in X, symbolized by t. or tx, which is {(x, x)lx C X}. For x, y in X, clearly, xtxy if x = y. If p is a relation
and A is a set, then p[A] is defined by p[A] = {yj forsome xinA,xpy}. 1.6 1 Relations 27 This set is suggestively called the set of p-relatives of elements of A. Clearly, p [Do J = R and, if A is any set, p [A 19 R. EXAMPLES 6.7. If Y 0 0, then Dxxy = X, and if X 0 0, then Rxxr = Y. 6.8. The basis for plane analytic geometry is the assumption that the points of the Euclidean plane can be paired with the members of It X R, the set of ordered pairs of real numbers. Thereby the study of plane geometric configurations may be replaced by that of subsets of R X K, that is, relations in R. For geometric configurations which are likely to be of interest, one can anticipate that the defining property of the associated relation in R will be an algebraic equation in x and y, or an inequality involving x and y, or some combination of equations and inequalities. In this event it is standard practice to take the defining property of the relation associated with a configuration as a description of the configuration and omit any explicit mention of the relation. For example, "the line with equation y = 2x + 1" is shorthand for "the set of points which are associated with {(x, y) C R X Rly = 2x + 1)." Again, "the region defined by y < x" is intended to refer to the set of points associated with {(x, y) C R X R{y < x}. As a further example, x <0andy> Dandy <2x-1-1 serves as a definition of a triangle-shaped region in the plane, as the reader can verify. If relations in It, instead of sets of points in the plane, are the primary objects of study, then the set of points corresponding to the members of a relation is called the graph of the relation (or of the defining property of the relation). Below appear four relations, and above each is sketched its graph. When the graph includes a region of the plane, this is indicated by shading. y y x {(x,y)CRXRly=.r} Figure 7 {(x,y)CRXRIy>_x} Figure 8 28 y Sets and Relations i CHAP. I y
0 x ((x,y)CRxRIO<x<2or 0<y<1) Figure 9 {(x,y)CRxRIO<x<2and 0<y<1) Figure 10 If p is the relation in R with 0 < x < 2 as defining property and o is the relation in B. with 0 < y < I as defining property, then the relation accompanying Figure 9 is equal to p U cr, and the relation accompanying Figure 10 is p n a. 'T'hus, Figures 9 and 10 illustrate the remarks that the graph of the union of two relations, p and o, is the union of the graph of p and the graph of o, and the graph of p n o is the intersection of the graphs of p and o. 6.9. Let p be the relation "is the father of." If A is the set of all men now living in the United States, then p[A] is the set of all people whose fathers now live in the United States. If A = (Adam, Eve), then p[A] = (Cain, Abel). EXERCISES 6.1. Show that if (x, y, z) _ (zt, v, zu), then x = u, y - a, and z = w. 6.2. Write the members of 11, 21 X (2, 3, 4). What are the domain and range of this relation? What is its graph? 6.3. State the domain and the range of each of the following relations, and then draw its graph. (a) {(x,y)CRXRIx2+4y2= 1}. (b) (c) {(x,y)CRXRI IxI+21yj= 1}. (d) {(x,y)CR XRIx2+y2 <1 and x> 01. (e) {(x,y)CRxB.Iy> Oandy <xandx+y < 1}, 6.4. Write the relation in Exercise 6.3(c) as the union of four relations and that in Exercise 6.3(e) as the intersection of three relations. 6.5. The formation of the cartesian product of two sets is a binary operation for sets. Show by examples that this operation is neither commutative nor associative. 6.6. Let Q be the
relation "is a brbther of," and let o be the relation "is a sister of." Describe 0 U a,,# n o, and 6 - o. 1.7 1 Equivalence Relations 29 6.7. Let a and v have the same meaning as in Exercise G.G. Let A be the set of students now in the reader's school. What is /3[A]? What is (0 U (r)[A]? 6.8. Prove that if A, B, C, and D are sets, then (.l fl B) X (C fl D) (A X C) n (B X D). Deduce that the cartesian multiplication of sets disthat (A fl B) X C = tributes over the operation of intersection, that (AXC)f1(BXC) and AX(BfC)=(AXB)fl(AXC) for all A, B, and C. is, 6.9. Exhibit four sets A, B, C, and 1.) for which (A U B) X (C U D) 5A (A X C) U (B X D). 6.10. In spite of the result in the preceding exercise, Cartesian multiplication distributes over the operation of union. Prove this. 6.11. Investigate whether union and intersection distribute over cartesian multiplication. 6.12. Prove that if A, B, and C are sets such that A 5-' 0, 13 (AXB)U(BXA)=CXC,then A- 13=C. 0, and 7. Equivalence Relations A relation p in a set X is reflexive (in X) iff xpx for each x in X. II no set X is specified, we assume that X = Dp U R. A relation p is symmetric if xpy iinpliesypx, and it is transitive iff xpy acrd ypz imply xpz. Relations having these three properties occur so frequently in rnatheutatics they have acquired a narrre. A relation p in X is an equivalence relation (in X) if p is reflexive (in X), symmetric, and transitive. If a relation p in X is an equivalence relation in X, then Dp = X. Because of this we shall henceforth use the terminology "an equivalence relation on X" in place
of "an equivalence relation in X." EXAMPLES Each of the following relations is an equivalence relation on the accompany- ing set. 7.1. Equality in a collection of sets. 7.2. The geometric notion of similarity in the set of all triangles of the Euclidean plane. 7.3. The relation of congruence modulo it in Z. This relation is defined for a nonzero integer n as follows: x is congruent to y, symbolized x - y(mod n), if n divides x - y. 7.4. The relation c\) in the set of all ordered pairs of positive integers where (x, y) cv (u, v) iff xv = yu. 7.5. The relation of parallelism in the set of lines in the Euclidean plane. 7.6. The relation of having the same number of members in a collection of finite sets. 30.Sets and Relations I CHAP. I 7.7. The relation of "living in the same house" in the set of people of the United States. The last example above illustrates, in familiar terms, the central feature of any equivalence relation: It divides the population into disjoint subsets, in this case the sets of people who live in the same house. Let us establish our contention in general. If p is an equivalence relation on the set X, then a subset A of X is an equivalence class (p-equivalence class) if there is a member x of A such that A is equal to the set of all y for which xpy. Thus, A is an equivalence class iff there exists an x in X such that A = p[(x[ 1. If there is no ambiguity about the relation at hand, the set of all p-relatives of x in X will be abbreviated [x] and called the equivalence class generated by x. Two basic properties of equivalence classes are the following. (I) X E [xI. (II) if xpy, then [x] _ (y]. The first is a consequence of the reflexivity of an equivalence relation. To prove the second, assume that xpy. Then [yj C [x] since z E [y] (which means that ypz) together with xpy and the transitivity of p yield xpz or z C [x]. The symmetry of p may
be used to conclude the reverse inclusion, and the equality of [x] and [y] follows. Now property (I) implies that each member of X is a member of an equivalence class, and (II) implies that two equivalence classes are either disjoint or equal since if z E [xl and z C [y], then [x] = [z], [yI = [z], and hence [x] = [yl. Recalling the definition of a partition of a nonempty set, we conclude that the collection of distinct p-equivalence classes is a partition of X. This proves the first assertion in the following theorem. THEOREM 7.1. Let p be an equivalence relation on X. Then the collection of distinct p-equivalence classes is a partition of X. Conversely, if 6' is a partition of X, and a relation p is defined by apb if there exists A in 6' such that a, b C A, then p is an equivalence relation on X. Moreover, if an equivalence relation p determines the partition (P of X, then the equivalence relation defined by 6' is equal to p. Conversely, if a partition 6' of X determines the equivalence relation p, then the partition of X defined by p is equal to 6'. Proof. To prove the second statement, let (P be a partition of X. The relation p which is proposed is symmetric from its definition. If a C X, there exists A in CP with a C A, so that p is reflexive. To 1.7 ( Equivalence Relations 31 show the transitivity of p, assume that apb and bpc. Then there exists A in 6' with a, b C A, and there exists B in 6' with b, c C B. Since b C A and b E. B, A = B. Hence apc. To prove the next assertion, assume that an equivalence relation p on X is given, that it determines the partition 6' of X and, finally, that 6' determines the equivalence relation p. We show that p = p. Assume that (x, y) C p. Then x, y C [xI and [x] C 6'. By virtue of the definition of p* it follows that xpy or (x, y) C p. Conversely, given (x, y) C p*, there exists A in 6
' with x, y C A. But A is a p-equivalence class, and hence xpy or (x, y) C p. Thus, p = p*. The last part of the theorem is left as an exercise. To illustrate part of the above theorem let us examine the equivalence relation of congruence modulo n on `_Z which was defined in Example 7.3. An equivalence class consists of all numbers a + kit with k, [n - I ] are distinct classes. 'T'here in Z. Clearly, therefore, [0], 111, are no others, since any integer a can be written in the form a = qn A- r, 0 < r < n, and hence a C [r]. A class of congruent numbers is often called a residue class modulo n. The collection of residue classes modulo n will be denoted by Z,.. We can use this example to emphasize the fact that, for any equivalence relation p, an equivalence class is defined by any one of its members, since if xpy, then [x] = [y]. Thus, [0 ] _ [n I = [2n], and so on, and [1 ] = [n + 11 _ [1 - n], and so on. If p is an equivalence relation on X, we shall denote the partition of X induced by p by X/p (read "X modulo p") and call it the quotient set of X by p. The significance of the partition of a set X accompanying an arbitrary equivalence relation p on X is best realized by comparing p with the extreme equivalence relation on X of identity. We classify identity on X as an extreme equivalence relation because the only element equal to a given element is itself. That is, the partition of X determined by identity is the finest possible-the equivalence class generated by x consists of x alone. In contrast, for two elements to be p-equivalent they Must merely have a single likeness in common, namely, that characterized by p. A p-equivalence class consists of all elements of X which are indiscernible with respect to p. That is, an arbitrary equivalence relation on X defines a generalized form of equality on X. On turning from the elements of X to the p-equivalence classes we have the effect of identifying any two elements which are p-equivalent. If p happens to preserve various structural
features of X (assuming it has such), these may appear in simplified form in X/p because of the identification of 32 Sets and Relations I CIIAP. 1 elements which accompanies the transition to X/p. Examples of this arise quite naturally later. Among the applications of equivalence relations in mathematics is that of formalizing mathematical notions or, as one often says, forrnulating definitions by abstraction. The essence of this technique is defining a notion as the set of all objects which one intends to have qualify for the notion. This seems incestuous on the surface, but in practice it serves very nicely. For example, let us consider the problem of defining the positive rational numbers in terms of the positive integers. Instead of defining ratios of integers directly we introduce the notion of pairs of integers having equal ratios by the definition (x, y) N (u, v) if xv = yu. This is an equivalence relation on _L+ X Z"{-, and we-can now define a rational number as an equivalence class. That is, the notion of equivalence of pairs of integers amounts to imposing a criteria for indiscernibility on "L_f- X Z+. Since this is an equivalence relation, a partition of the universe of discourse is at hand, and in an equivalence class we have the abstraction of the property common to all of its members. Thus we define a rational number to be such an equivalence class. The familiar symbol x/y emerges as an abbreviation for the equivalence class [(x, y)]. That an equivalence class is defined by each of its members implies that any other symbol u/v, where (u, v) C [(x, y) ], may be taken as a name for the same rational number. For example, the statement 2/3 = 4/6 is true because 2/3 and 4/6 are merely different names for the same rational number. Another instance of definition by abstraction is that of direction based on the equivalence relation of parallelism: a direction is an equivalence class of parallel rays. The notion of shape may be conceived in a like fashion: geometric similarity is an equivalence relation on the set of figures in the Euclidean plane, and a shape may be defined as an equivalence class under similarity. So far, the fundamental result concerning an equivalence relation pthat the collection of all distinct p-equivalence classes is disjoint and xpy if x and y are members of the san
ic equivalence class-has been employed solely in connection with applications of equivalence relations. It can also be made the basis of a characterization of equivalence relations among relations in general. This is clone next. THEOREM 7.2. A relation p is an equivalence relation if there exists a disjoint collection 41 of nonempty sets such that p = { (x, y)l for some C in P, (x, y) C C X C). 1.7 I Equivalence Relations 33 Proof. Assume that p is an equivalence relation on X. Then the collection of distinct p-equivalence classes is disjoint, and we contend that with this choice for (P, p has the structure described in the theorem. We show first that { (x, y) I for some C in (P, (x, y) C C X C) C p. Assume that (x, y) is a member of the set on the left side of the inclusion sign. Then there exists an equivalence class [z] with x, y E [z]. Then zpx and zpy, and hence xpy, which means that (x, y) C p. To show the reverse inclusion, assume that (x, y) C p. Then x, y C [x], and hence (x, y) C [x] X [x]. The proof of the converse is straightforward and is left as an exercise. EXERCISES 7.1. If p is a relation in R', then its graph is a set of points in the first quadrant of a coordinate plane. What is the characteristic feature of such a graph if: (a) p is reflexive, (b) p is symmetric, (c) p is transitive? 7.2. Using the results of Exercise 7.1, try to formulate a compact character- ization of the graph of an equivalence relation on R+. 7.3. the collection of sets {{1, 3, 4}, (5, 6}} is a partition of 11, 2, 3, 4, 5, 6, 7). Draw the graph of the accompanying equivalence relation. 7.4. Let p and v be equivalence relations. Prove that p f v is an equivalence (2, 7}, relation. 7.5. Let p be an equivalence relation on X and let Y be a set. Show that P n (Y
x Y) is an equivalence relation on X (1 Y. 7.6. Give an example of these relations. (a) A relation which is reflexive and symmetric but not transitive. (b) A relation which is reflexive and transitive but not symmetric. (c) A relation which is symmetric and transitive but not reflexive in some set. 7.7. Complete the proof of Theorem 7.1. 7.8. Each equivalence relation on a set X defines a partition of X according to Theorem 7.1. What equivalence yields the fittest partition? the coarsest partition? 7.9. Complete the proof of "I'Ircorem 7.2. 7.10. Let p be a relation which is reflexive and transitive in the set A. For a, b C A, define a N b if apb and bpa. (a) Show that - is an equivalence relation on A. (b) For [a], [b.l E Al-, define [a]p'[b] if apb. Show that this definition is independent of a and b in the sense that if a' C [a], b' C [b], and apb, then a'pb'. (c) Show that p' is reflexive and transitive. Further, show that if [a]p'[b] and [b]p'[a], then [a] = [b]. 7.11. In the set Z+ X Z+ define (a, b) - (c, d) if a + d = h + c. Show that 34 cu is an equivalence relation on this set. Indicate the graph of Z+ X Z_+, and describe the ca-equivalence classes. Sets and Relations I CHAP. 1 8. Functions It is possible to define the concept of function in terms of notions already introduced. Such a definition is based on the common part of the discussions about functions to be found in many elementary texts, namely, the definition of the graph of a function as a set of ordered pairs. Once it is recognized that there is no information about a function which cannot be derived from its graph, there is no need to distinguish between a function and its graph. As such, it is reasonable to base a definition on just that feature of a set of ordered pairs which would qualify it to be a graph of a. function. This
we do by agreeing that a function is a relation such that no two distinct members have the same first coordinate. 't'hus, f is a function ifT it meets the following requirements. (1) The members off are ordered pairs. (II) If (x, y) and (x, z) are members of f, then y = z. EXAMPLES 8.1. {(1, 2), (2, 2), (Roosevelt, Churchill)) {1, 2, Roosevelt) and range (2, Churchill). is a function with domain 8.2. The relation {(1, 2), (1, 3), (2, 2)) is not a function, since the distinct members (1, 2) and (1, 3) have the same first coordinate. 8.3. The relation {(x, x2 + x + 1)Jx E It} is a function, because if x = u, then x2-l-x+1 = u2-- u-1-1. 8.4. The relation {(x2, x)Jx C It} is not a function, because both (1, 1) and (1, -1) are members. Synonyms for the word "function" are numerous and include transformation, map or mapping, correspondence, and operator. If f is a function and (x, y) E f, so that xfy, then x is an argument of f. There is a great variety of terminology for y; for example, the value off at x, the image of x under f, the element into which f carries x. There are also various symbols for y: xf, f(x) (or, more simply, fx), xf. The notation "f(x)" is a name for the sole member off! {x} ], the set off-relatives of x. In these terms the characteristic feature of a function among relations in general is that each member of the domain of a function has a single relative. The student must accustom himself to these various notations, since 1.8 I Funclions 35 he will find that all are used. In this book definitions and theorems pertaining to functions will consistently be phrased using the notation f(x), or fx, for the (unique) correspondent of x in a function f. The notation f[AJ for {yj for sonic x in A,
(x, y) C f) is in harmony with this. However, in applications of functions we shall use a variety of notations. When it is more convenient to use xf in place of f(x), then [All will be used in place off [A]. If xf is used in place of f(x), then [A if or Af will be used in place off [A I. Since functions are sets, the definition of equality of functions is at hand: Two functions f and g are equal if they have the same members. It is clear that this may be rephrased f = g iff Df = Dv and f(x) = g(x) for each x in the common domain. Consequently, a function may be defined by specifying its domain and the value of the function at each member of its domain. The second part of this type of definition is, then, in the nature of a rule. For example, an alternative definition of the function { (x, x2 + x -}- I )jx C R) is the function f with B. as domain and such that f(x) = x1 + x + 1. When a function is defined by specifying its domain and its value at each member of the domain, the range of the function may not be evident. The above example requires a computation to conclude that Rf = {x C RJx > 11 ). On 4 the other hand, it is almost obvious that Rf C R+-. In general, one can anticipate difficulty in determining the range, but no difficulty in determining some set that includes the range. Thus, it is convenient to have available the following terminology. A function f is into Y if the range of f is a subset of Y, and f is onto Y if Rf = Y. For corresponding notation for the domain of a function we shall say that f is on X when the domain off is X. The symbols f: X-*- Y and X-f3.- Y are commonly used to signify that f is a function on the set X into the set Y. The set of all functions on X into Y, symbolized YX, is a subset of 61(X X Y). If X is empty, then YX consists of only one member-the empty subset of X X Y. This is the only subset of X X Y, since when X is empty so is X X Y. If Y is empty and X
is nonempty, then Yx is empty. In summary, Yo = 101 36 Sets and Relations I CHAP. 1 and Ox=0 if X 0. If f : X -+ Y, and if A C X, then f n (A X Y) is a function on A into Y (called the restriction of f to A and abbreviated fIA). Explicitly, f JA is the function on A such that (f lA) (a) = J(a) for a in A. A function g is the restriction of a function f to some subset of the domain off iff the domain of g is a subset of the domain of f and g(x) = f(x) for x C D,,; in other words, g C f. Complementary to the definition of a restriction, the function f is an extension of a function g iff g C f. In order to present an example of the notion of a restriction of a function we recall the earlier definition of the identity relation tx in X. Clearly, this relation is a function, and hence, in keeping with our current designation of function by lower-case English letters, we shall designate it by i or ix. We shall call ix the identity map on X. If A C X, then ixI A = in If ixJA is considered as a function on A into X, then it is the injection mapping on A into X. A function is called one-to-one if it maps distinct elements onto dis- tinct elements. That is, a function f is one-to-one if xr ; x2 implies f (XI) 0 f (x2). In demonstrating one-to-oneness it may prove to be more convenient to use the contrapositive of the foregoing: implies f(XI) = f(x2) x1 = x2. For example, the function f on R such that J(x) = 2x + I is one-to-one since 2xr + 1 = 2x2 -I- I implies xr = x2. If f is a one-to-one function on X onto Y or, somewhat less awkwardly, if f : X --} Y is one-to-one and onto, then it effects a pairing of the elements of X with those of Y upon matching f (x) in Y with x in X. Indeed, since f is a function, f
(x) is a uniquely determined element of Y; since f is onto Y, each y in Y is matched with some x; and since f is one-to-one, each y is matched with only one x. Because of the symmetrical situation that a one-to-one map on X onto Y portrays, it is often called a one-to-one correspondence between X and Y. Also, two sets so related by some function are said to l)c in one-to-one correspondence. EXAMPLES 8.5. The familiar exponential function is a function on R into R, symbolized f : R -+- ] with f(x) = eE. 1.8 I Functions 37 We can also say, more precisely, that f is a function on R. onto R{. In general, if J: X -} Y, then f is a function on X onto f [X], that is, onto the range of f. B.G. {a, b, c} (r. 2) is the set of all functions on (1, 2) into (a, b, c}. One mem- ber of this set is {(1, a), (2, c)). 8.7. If A and B are sets having the same number of elements, they clearly are in one-to-one correspondence. Then it is an easy matter to show that for any set X, Ax and Bx are in one-to-one correspondence. This being the case, it is customary to denote the set of all functions on X into any set of n elements by nx. Thus, 2x denotes the set of all functions on X into a set of two elements, which we will ordinarily take to be (0, 11. If A C X, then one member of 2X is the function XA defined as XA(x) = I if x C A, and XA(x) = 0 if x C X - A. We call XA the characteristic function of A. Now let us define a function f on 6'(X) into 2x by taking as the image of a subset A of X [that is, a member of 6'(X)] the characteristic function of A (which is a member of 2x). It is left as an exercise to prove that f is a one-to-one correspondence between (P(X) and 2x. It is customary to regard (
P(X) and 2x as identified by virtue of this one-to-one correspondence, that is, to feel free to replace one set by the other when it is convenient. 8.8. If f is a function and A and B are sets, then it can be proved that f[A U B] = J[A] U f[B] and that J[A n B] s f[.A] n f[B]. The inclusion relation in the case of A n B cannot be strengthened. In elementary mathematics one has occasion to use functions of several variables. Within the frhrncwork of our discussion a function of n variables (n > 2) is simply a function whose arguments are ordered n-tuplcs. We can include the case it = 1 if we agree that a I-tuplc, (x), is simply x. Introducing the notation X" for the set of all n-tuplcs, x"), where each x is a member of the set X, a function, (x,, x2, whose domain is X" and whose range is included in X, is an n-ary operation in X. In place of "l-ary" we shall say "urrary"; for example, complementation is a unary operation in a power set. In place of "2-ary" we shall say "binary." This was anticipated in our discussion of operations for sets; for example, into rsection is a binary operation in a suitable collection of sets. Also, addition in Z is a binary operation; if x, y C Z, the value of this function at (x, y) is written x + y. EXERCISES 8.1. Give an example of a function on R onto Z. 8.2. Show that if A C X, then ixIA = iA. 8.3. If X and Y arc sets of n and n: elements, respectively, Yx has how many elements? How many members of 6'(X X Y) are functions? 38 Sels and Relations I e tt A P. 1 8.4. Using only mappings of the form f: Z.t --'- Z4', give an example of a function which (a) is one-to-one but not onto; (b) is onto but not one-to-one. 9.5. Let A = {1, 2,, n). Prove that
if a map f: A -3- A is onto, then it is one-to-one, and that if a map g: A A is one-to-one, then it is onto. 8.6. Let f : R+ --} R, where f(x) = fy dl' Show as best you can that f is a o ne-to-one and onto function. 8.7. Prove that the function f defined in Example 8.7 is a one-to-one corre- spondence between W(X) and 2x. 8.8. Referring to Example 8.8, prove that if f is a function and A and B are sets, then f[A U B] = f[A] U f[B]. 8.9. Referring to the preceding exercise, prove further that J[A n B] C J[A] n f [B], and show that proper inclusion can occur. 8.10. Prove that a function J is one-to-one if for all sets A and B, f[A fl B] f[A] n ffB]. 8.11. Prove that a function f : X -} Y is onto Y iff f [X - A] Q Y - f[A] for all sets A. 8.12. Prove that a function f: X - )- Y is one-to-one and onto iff f [X - A] Y - f [A] for all sets A. 9. Composition and Inversion for Functions To motivate our next definition, we consider an example. Let the functions f and g be defined as f: R -4- B. with f(x) = 2x + 1, g: RF-}Rf with g(x) = x1'2. It is a familiar experience to derive from such a pair of functions a function h for which h(x) = g(f(x)). Since the domain of g is Rt- by definition, x trust be restricted to real numbers such that 2x + I > 0 for h(x) to be defined. That is, combining f and g in this way yields a function whose domain is the set of real numbers greater than -1 and whose value at x is g(J(x)) = (2x + 1)1/2 The basic idea of this example is incorporated in
the following definition. By using ordered pair notation (instead of the domain and value no(ation) for functions, we avoid having to make any restriction stermning from a difference between the range off and the domain of g. The composite of functions f and g, symbolized g ° f, 1.9 I Composition and Inversion for Functions 39 is the set (x, z)l there is a y such that xfy and ygz 1. It is left to the reader to prove that this relation is a function. This operation for functions is called (functional) composition. The following special case of our definition is worthy of note. If then f: X -}} andg:Y -i- Z, g -f: X --3- /, and (g ° f)(x) = g(f(x)) The above example establishes the fact that functional composition is not a commutative operation; indeed, rarely does f o g = g o f. however, composition is an associative operation. That is, if f, g, and h are functions, then f°(goh) = (fog)oh. To prove this, assume that (x, u) C f o (g - It). 'T'hen there exists a z such that (x, z) C g o h and (z, a) C f. Since (x, z) C g o h, there exists a y such that (x, y) C h and (y, z) C g. Now (y, z) C g and (z, u) C f imply that (y, u) C f - g. Further, (x, y) C It and (y, u) C f o g imply that (x, u) C (f o g) o Is. Reversing the foregoing steps yields the reverse inclusion and hence equality. The foregoing proof will be less opaque to the reader if he rewrites it in terms of function values. The proof given is in accordance with our definition of functional composition and has the merit that it avoids any complications arising from a difference between the range off and the domain of g. From the associative law for composition follows the general associative law, which the reader may formulate. The unique function which is defined by composition from the functionsf,, fz, - - -, f, in that order will be designated by fl 'f2 0... 'J.- EX
AMPLES 9.1. Let h: It where h(x) _ (1 -l- x2)e2. Then /t = g - f if f : R with f(x) = 1 -1 x2, and g: R+ -- R+ with g(x) = x'12. It is this decomposition of /i which is used in computing its derivative. 9.2. A decomposition of an arbitrary function along somewhat different lines than that suggested by the preceding example call be given in terms of concepts we have discussed. First we make a definition. If p is an equivalence relation with domain X, then j: X -- X/p with j(x) = (A) 40 is onto the quotient set X/p; j is called the canonical or natural mapping on X onto X/p. Now, if f is a mapping on X into Y, the relation defined by Sets and Relations I C IL A P. I XJPX2 ifl'f(x,) = f(x2) is clearly an equivalence relation on X. Let j be the canonical map on X onto X/p. We contend that a function g on X/p into f[X], the range of f, is defined by setting g([x]) = f(x). To prove that g is a function, it must be shown that if [x] _ [Y] then f(x) = f(y). But [x] _ [y] if xpy if f(x) = f(y); so g is a function. Finally, we let i be the injection of f[X] into Y. Collectively, we have defined three functions j, g, i where j: X-' X/p with J(x) = [x], g: X/p-} f[X] with g([x]) = I(x), i: f[X] -} Y with i(y) = y. Clearly, j is onto and i is one-to-one. It is left as an exercise to show that g is one-to-one and onto and that f=iog"1. 'I this equation is the whole point of the discussion. It proves to be a useful decomposition for an arbitrary function f. 9.3. If f is a known function with domain X and with range a subset of Y, then the notation f
: X-- Y for f includes superfluous information. However, it does suggest the consideration of f as a function that is associated with the pair (X, Y) of sets X and Y. If g: Y-+ Z is likewise associated with (Y, Z), then we associate the composite function g -f with (X, Z). The association of each function f with a pair of sets X and Y, such that X is the domain of f and Y includes the range of f and the agreement that the composite g e f of f : X --} Y and g: W - Z may be formed only if W = Y, has certain merits. For example, within this framework it is possible to characterize "onto" (along with "oneto-one") as a property of functions. Further, it sets these forth as dual properties in a sense that will be explained later. The characterization of one-to-oneness that we can demonstrate is as follows. (I) I.ct f : X -- Y. Then f is one-to-one ifT for all functions g and h such that g: Z -- X and h: Z -} X, fog = f o h implies that g = h. Indeed, suppose that f is one-to-one and that g and h are mappings on Z into X for which f - g = f ^ h. Then f(g(z)) = f(h(z)) for all z in Z. With f oneto-one it follows thatg(z) = h(z) for all z in Z. Hence, g = h. The proof of the converse is left as an exercise. A characterization of a function being "onto" can now be given by a simple alteration of M. (11) Let f : X -} Y. Then f is onto Y ifT for all functions g and h such that g: Y -} 7 and h: 1' -} Z, g o f = h o f implies g = h. The proof is left as an exercise. With the above characterizations at our disposal the decomposition obtained in Example 9.2 can be described more neatly as follows. For any function f 1.9 I Composition and Inversion for Functions 41 there exists a function i which is one-to-one, a function j which is onto, and a function g which is one-to-one and onto, such that f = i o g
- j. If the coordinates of each member of a function f (considered as a set of ordered pairs) are interchanged, the result is a relation g which may not be a function. Indeed, g is a function if (y, x) and (y, z) in g imply that x = z. In terms of f this means that if (x, y) and (z, y) are in f, then x = z, that is, f is one-to-one. If f is one-to-one, the function resulting from f by interchanging the coordinates of members of f is called the inverse function of f, symbolized i This operation, which is defined only for one-to-one functions, is called (func(ional) inversion. If f_.., exists, then its domain is the range off, its range is the domain of f, and x = `(y) if y = f(x). Further, f--' is is equal to f. If f is a one-to-one one-to-one and its inverse, function on X onto Y, then f' is a one-to-one function on Y onto X. Moreover, f-'of = ix, and fof-' = iv. There is another important connection between composition and inversion of functions. If f and g are both one-to-one functions, then g o f is one-to-one, and The proof is left as an exercise. (g ° f) -' = f -' ° g--'. EXAMPLES 9.4. The function f: R ->- R such that f(x) = 2x + 1 is one-to-one. The inverse off may be written {(2x -I- 1, x)Ix C R}. This is not very satisfying to one who prefers to have a function defined in terms of its domain and its value at each member of the domain. To satisfy this preference, we note that {(2x -1- 1, x)Jx C I_l} = {(t, a(t - 1))ft C R}. Thus ff' is the function on R into R such that f' '(x) = J(x - 1). 9.5. The function g: R'- 1Z} such that g(x) = x2 is one-to-one, since xi
= x22, and both x, and x2 positive imply that x, = x2. Then g'': R-'' ---r R' where g-'(x) = x112. 9.6. The function f : R -+ R'- where f (x) = 100 is known to be one-to-one and onto. The inverse function is called the logarithm function to the base 10, and its value at x is written logo x. The equations 42 Sets and Relations I CHAP. 1 log,o10,=x,forxCR,and10"' = x, forx>0, are instances of equations (f'o f)(x) = x, for x C D1, and (f of ')(x) = x, for x C R1, which are true for any one-to-one function. 9.7. If the inverse of a function f in It exists, then the graph of f'may be obtained from that off by reflection in the line y = x. The proof is left as an exercise. if the inverse of a function f is defined, then j '[A U B] = f '[A] U PI[B] and f '[A n B] c f '[A] o f '[B]. The latter identity can be sharpened to f-'[A n B] = f-'[A] n f -'[B] for inverse functions. The proof is left as an exercise. A set of the form f-'[A] we call the inverse or counter image of A under f. 9.8. From Example 8.8, EXERCISES 9.1. Let J: R -+- R where f(x) = (1 - (1 - x)113)"5. Express f as the com- posite of four functions, none of which is the identity function. 9.2. Iff:X --}Yand A X, show that/IA = f - iA. 9.3. Complete the proof of the assertions made in Example 9.2. 9.4. Complete the proof of (I) and supply a proof of (11) in Example 9.3. 9.5. Prove that f : A -+- B is a one-to-one correspondence between A and B if there exists a map g: B -* A such that g e f = iA and f
o g - iB. 9.6. Iff : A ->- B and g: B -'- C are both one-to-one and onto, show that g -f: A -a- C is one-to-one and onto and that (g -f)-1 = f'o g-1. 9.7. For a function f: A - A, fn is the standard abbreviation for f of o -f with n occurrences of f. Suppose that fn = i4. Show that f is one-to-one and onto. 9.8. Justify the following restatement of Theorem 7.1. Let X be a set. Then there exists a one-to-one correspondence between the equivalence relations on X and the partitions of X. 9.9. Prove that if the inverse of the function f in R exists, then the graph of may be obtained from that off by a reflection iu the line y = x. 9.10. Show that each of the following functions has an inverse. Determine the domain of each inverse and its value at each member of its domain. Further, sketch the graph of each inverse. (a) f: R -' B wheref(x) - 2x - 1. (b) f : B. -' It where f (x) = x'r. (c) f = {(x, (1 - x2)'/2)J0 < x 11}. (d) f =l(x,x X. 1)I-2<x<1}. 9.11. Establish identity (g -f)-'J=f-' -g-1 for one-to-one functions f and g. 9.12. Prove that if the inverse off exists, then f '[A n B] = f-'[A] n f"-'[B]. 1. 10 I Operations for Collections of Sets 43 9.13. The definition of the composite of two functions is applicable to any pair of relations. With this in mind, show that if f is any function and g = {(y, x)f (x, y) C f} then g o f is an equivalence relation. 9.14. Let A, B, A', and B' be sets such that A and A' are in one-to-one correspondence and B and B' are in one-to-one correspondence.
Show that (a) there exists a one-to-one correspondence between A X B and A' X B'; (b) there exists a one-to-one correspondence between AB and A'B (c) if, further, A fl B = 0 and A' n B' = 0, then there exists a one-toone correspondence between A U B and A' U If'. 9.15. For sets A, B, and C show that (a) A X B is in one-to-one correspondence with B X A; (b) (A X B) X C is in one-to-one correspondence with A X (B X C); (c) A X (B U C) is in one-to-one correspondence with (A X B) U (A X C). 9.16. For sets A, B, and C show that (a) (A X B)c is in one-to-one correspondence with Ac X Be; (b) (AB)C is in one-to-one correspondence with Arrxc; (c) if, further, B n C = 0, then ABuc' is in one-to-one correspondence with As X A(,'. 10. Operations for Collections of Sets In this section we generalize the binary operations of union, intersec- tion, and Cartesian product. Let (I, be a collection of sets. The union of (t is the set of all objects x such that x belongs to at least one set of the collection ct. That is, it is (xlx C X for some X in (t}. This set is symbolized by U(t or U(XIX E a,} or U,-,X. The earlier definition of A U B is seen to be simply the union of (A, R}. That is, U(XIXE(A,B}}=AUB. In Section 5, using the property of associativity of union as a binary operation, we defined what is immediately seen to be in our present, terminology the union of a collection of the type {A,, A2, We. U A. for this union. shall continue to use the denotation A, U A2 U From the viewpoint of set theory, it was a waste of space to have introduced this extension. However, from the viewpoint of the algebra of sets, it was not. 44 EXAMPLES Se/.
s and Relations! C H A P. 1 10.1. UO=0. 10.2. U {A} = A. 10.3. If a _ ({1, 2}, {a., {(1, 2)}, {(3, 4) }, 0) and UP(a) = {{1, 21, {3, 4)) = Lt. It is left as an exercise to show that U(P(a) - (% is an identity. {3, 4}}, then U(1. _ (1, 2, 3, 4}. Also, (P(CB) The intersection of a nonempty collection a of sets is the set of all objects x such that x belongs to every set of the collections a. That is, it is (xjxCXforallXina}. This set is symbolized) by na or n(XIXC a} or n.vcr,X. Earlier, A n 13 was defined as the intersection of (A, 13}. That is, n(xixc (A, 13}} = An B. Further, the. earlier definition of A, n A2 n... n A,, coincides with, A J. what we may now call the intersection of the collection (At, A2, The question of why the definition of the intersection of a collection of sets has been restricted to nonempty collections deserves an answer. if the defining property for the intersection is applied to the empty collection, we have no = (xjx C X for all X in o }. It is left to the reader to convince himself that the defining property at hand is satisfied by any object whatsoever. Clearly this is an unsatisfactory situation. An alternative which may be offered is based upon the assumption that there is a universal set U at hand. '1'lrcn the intersection of a collection 0, (of subsets of U) is defined to be (x C Ulx C.I for all X in Ct}. For a nonempty collection, the new definition agrees with the old. The cliffercncc is the way in which they treat the empty collection; according to the new definition, n xcox = U, which seems to be a more reasonable result. Algebraic properties of unions and intersections will he presented in terms of one of the standard notations for designating collections of sets. In this notation, a collection of sets. appears as the range of a suitable 1
.10 I OJperi Lions for Collections of Sets 45 function. To this end we introduce sorne definitions. Suppose that y is a function on a set I into a set Y. Let us call an clement i of the domain I an index, I itself an index set, the range of y an indexed set, and the function y itself a family. We shall denote the value of y at i by yi and call y, the ith coordinate of the family. Thereby, we may write y= {(i, yi) C I X Vii C I}. Actually, y is completely specified by { y4I i C 11 ; in this notation it is the range of the function which is emphasized. In place of" (yi;i C 11,11 it is common practice to write "{y,} with i E I" or, if the domain is clear from the context, simply "{yi}." Such notation has its origin in that employed for sequences. By definition, a sequence is a family on the set of positive (or, nonnegative) integers into a set Y. That is, a, R, } sequence is a function for which {1, 2, -, n, } or 10, 1, serves as an index set. Hereafter we shall denote the latter set by N. By the phrase "a family (Ail of subsets of II" we shall understand a function A on some set I of indices into o'(U). The union of the range of such a family is called the union of the family (Ail or the union of the sets A. The standard notation for it is U {A4;i E I} or UiE1Ai or U,A where the last denotation suggests that the index set need not be cmpliasized. For the ease of the union of a sequence { Ail i C Ni of sets A i, each of the notations U',--oAi and Ao U Ac U... U A. U..., is also used. Similarly, the union of (AI, A2, is denoted by U,'r;Ai or Ac U Az U U A,.. In every case it follows from the definition of unions that x E U;Ai iff__ x belongs to A, for at least one i. If we agree to use the second of the definitions given above for the intersection of a family {A, } of
subsets of 11, the terminology and notation for intersections parallel those for unions in every respect. 'T'hus, the intersection of the range of the family is called the intersection of the family [Ail or the intersection of the sets A,. The standard notation for this is n {Aji E I} or ni,,Ai If the family is nonempty, that is, if I 31- 0, then x E niAi iff x is a member of Ai for all i. If I = 0, then n,A, = U. or n,Ai. 46 Sets and Relations I C 11 A P. 1 Incidentally, it should be noted that there is no loss of generality in considering families of sets in place of arbitrary collections. As the reader can easily show, every collection of sets is the range of some family. In the following theorem appear several algebraic properties of unions and intersections of families; others appear among the exercises. These generalize properties of the operations of the operations of union and intersection for pairs. The reader may supply the proofs. TI 11: 0 R F M 10.1. Let (A,) with i C I be a family of subsets of U and let B C U. Then (I) B n U,A1 = U,(B n A,) and Bu n;A; = n;(B U A,). (II) U - U1A1 = n,(U - A,) and u- n,Ai = U,(U - A,). (III) If J is a subset of I, then U;c iAj C Ui(,Ai and niE-jAi Q 1 icrAi. EXAMPLES 10.4. In spite of the emphasis which has been given to the interpretation of a collection of sets as the range of some family, it should not be inferred that the accompanying notation is indispensable for stating results like those in Theorem 10.1. For example, the first distributive law in (1) may be stated for a collection tt. of subsets of U as B0 U(AIAE(t} =U{BnAIAE(t} and the first of the DeMorgan laws in (II) as U - U{AIAC(a} = n{U--AIAEt:I,}. 10.5. The following identities generalize those in Example 8
.8. Iff is a func- tion and {A,} is a family with nonempty domain I, then f[U,A,I = U,ffAil and f[n,A,l = n,f[A,l. Further, if f is one-to-one, then equality holds in the second identity (see Exercise 8.10). 10.6. The following compact formulation of Theorem 7.2 is now possible: A relation p is an equivalence relation ill' there exists a disjoint collection (3' of sets such that p - tc X CIC E o'}. We shall use the notion of a fancily to generalize the concept of the cartesian product of two sets. For this we note that an eleirient (a,, a1) of the cartesian product A, X A2 drones a family a with domain (1, 21 and whose values at 1 and 2 are a, C A, and a C A>, respectively. If A is the set of all families having 11, 2) as domain and such that their value at i is a member of A, for i = 1, 2, then the function f: A, X A2 --*- A, 1.10 1 Operations for Collections of Sets 47 where f(a,, a2) = a as described above, is a one-to-one correspondence. We take the existence of this one-to-one correspondence as the basis for the assertion that the only difference between A, X A2 and A is a notational one. As such, we shall henceforth not distinguish between them. The generalization of A, X A2, with A, X A2 regarded as A, is an easy matter. If (Ai} with i C I is a family of sets, then the cartesian product of the family, in symbols X { Aili C I } or XiE1Ai is the set of all families a with domain I and such that ai C Ai for each i in I. or XiA1, For the cartesian product of a sequence {Aili C NJ of sets Ai, the notation X1+O Ai or AoXA,X... XA.X... is used. Similarly, the cartesian product of (A,, A2, by X;s,Ai or A, X A2 X X A,,., is denoted As the latter symbol
isni suggests, if I = (1, 2 }, we shall identify Xic,Ai with A, X A2 as defined earlier and XieiAi with A, if I = (1 }. If every member of the f'ainily (A, } with i C I is equal to the same set X, then XiciAi = XI, the set of all functions on I into X. If I = (1, 2,., n J, then we identify Xi with X" as defined earlier. In particular, X' is taken to be X. We introduce one more bit of terminology for cartesian products. Let (Ai) with i C I be a fairrily of sets and let A be its cartesian product. If J is a subset of I, then there is a natural correspondence of the elements of A with those of XicjAi. To formulate this explicitly, we use the fact that an element a of A is a family {a,} with I as domain. Then the element b, let us say, of Xic,Ai which is the natural correspondent of a is the restriction of a to J. We shall write bi for a, when i C.1. The function on A whose value at a is b is called the projection on A onto XiErAi. If J = I j } and p, is the projection on A onto A;, then p;(a) is called the j-coordinate of a. EXERCISES 10.1. Let p be a relation, that is, a set each of whose members is an ordered pair. Show that p is a relation in UUp. 10.2. Show that ii' d is a collection of sets, then (a) a = U(P((t), arid (b) a P(Ua). 48 Sets and Relations I CHAP. I Can the inequality in (b) be strengthened? 10.3. Supply proofs for the identities in 'T'heorem 10.1. 10.4. Let {l,} be a family of sets with domain J. Let I - U;I; and suppose that {A,} is a family of sets with domain I. Prove the following associative laws. (a) UjCjAi (I)) ni(-[Ai = 10.5. Prove each of the distributive laws, (U,A1
) n (Uilli) = Ui.i(Ai n B1) and (niAi) U (n,B,) = ni.;(Ai u B;). Here it is to be understood that such a symbol as U,,; is an abbreviation for Ulii>E_tx 10.6. (a) If A and B are sets and X is (A, B), prove that UX = {A, B}, nx = (A}, U(nx) = A, n(nx) = A, U(UX) = A U 13, and n(Ux) _ AnB. (h) Suppose that it is known that the set.V is an ordered pair. Use the results in (a) to recapture the first coordinate and the second coordinate of X. 10.7. Prove that (U,A1) X (UjB,) = Ui,,(A, X 13,), as well as a like result for intersections. 10.8. Let {I,lj C.11 be a partition of the set 1. Determine a one-to-one corre- spondence between X,EIA1 and 11. Ordering Relations In this section we define several types of relations which have their origin in the intuitive notion of an ordering relation (order of preccdcncc), that is, a relation p such that for an appropriate set X there are various distinct members x and y of X such that xpy, but it is not the case that ypx. Then, by means of p, we could decide to put the x and y in question in the order x, y rather than y, x because xpy, and it is not the case that ypx. For a set of real numbers the familiar relations <, <, and > are used in this capacity. For a collection of sets the relations C and S serve similarly. The first ordering relation we shall consider has as its defining properties the basic features common to the above relations of < for numbers and C for sets. We define a relation p as antisymmetric if whenever xpy and ypx then x = y. A relation p in a set X is called a partial ordering (in X) if p is reflexive (in X), antisyininetric, and transitive: If no set X is specified we assume X = D, U Rp. For the consideration of
a partial ordering relation relative to various sets (for example, the familiar ordering in _I relative to the set of even integers), it is convenient 1.11 I Ordering Relations 49 to make the further definition that a relation p partially orders a set y itf p n (Y x Y) is a partial ordering in I'. The relation p f-1 (Y X Y) is the "restriction" of p to Y in the sense that it is reduced by all ordered pairs either of whose coordinates are not nicmbers of 1'. EXAMPLES 11.1. The relation "is an integral multiple of" in Z{ is a partial ordering. 11.2. A hierarchy or a table of organization in a business fine is determined by a partial ordering in some set of positions. 11.3. If p is a partial ordering in X, then p n (A X A) partially orders the subset A of X. 11.4. If p is a relation, the converse of p, symbolized by j,, is the relation such that yfix ifT xpy. If p is a partial ordering, then so is its converse. 11.5. A relation p that is reflexive and transitive is a. preordering. A potential shortcoming of such a relation, in connection with establishing an order of precedence in a set X, is the possibility of p being "indifferent" to some distinct pair x, y of objects in the. sense that both xpy and ypx. For example, in some population let w be the weight function and h be the height function of individuals so that rv(x) and h(x) arc the weight and height, respectively, of the individual named x. Then the relation p such that xpy if iv(x) < w(y) and h(x) < Iz(y) is a preordering, but is not a partial ordering if there are two iridividuals having the same weight and height. If p is a preordering in X, then it determines a partial ordering in a partition of X, according to Exercise 7.10. There it is asserted first that the relation cv such that x cv y iffi xpy and ypx is an equivalence relation. Secondly, it is stated that the relation p' such that [x] p' [_y] iff xpy is a partial ordering having the accompanying set of equivalent classes [x] as domain.
In summary, if p is a preordering in A, then it is a partial ordering in the set obtained from X by identifying elements to which it is indifferent. The foregoing is nicely illustrated by taking p as the relation in the set of complex numbers such that zpw if the real part of z is less than, or equal to, the real part of iv. We shall follow custom and designate partial orderings by the symbol <. If the relation < partially orders X, and x and y are members of X, it may or may not be the case that x < y. If it is not, we write y to x < y and say x is x less than y, or x precedes y, or y is greater than x. We shall also use y > x and y > x as alternatives for x < y and x < y, respectively, when it is coliveIlient. y. Also, we abbreviate x < y and x Defining a relation p in X as irrellexive (in X) if for no x in X is xpx, we see that if < is a partial ordering in X, then < is irreflexive 50 Sets and Relations I C FI A P. 1 and transitive in X. Conversely, starting with an irreflexive and transitive relation < in X, the relation < such that x < y iff' x < y or x = y is a partial ordering in X. The proofs are left as an exercise. The derivation of < from <, and vice versa, can be illustrated in concrete terms by the definition of proper inclusion for sets in terms of inclusion, and vice versa. If < partially orders the finite set X, the relation < can be expressed in terms of the following concept. An element y of X is a cover of x in X if x < y and there exists no u in X such that, x,, of X can be x < u < y. If x < y, then, clearly, elements x,, x2, found such that x = x, < x2 < < x = y, and each xj i., covers x,. Conversely, the existence of such a sequence implies that x < y. A relation p is a simple (or linear) ordering ifI it is a partial ordering such that xpy or ypx whenever x and y are distinct members of the domain (which is equal to the range) of p. A relation p simply orders a set
Y if p n (Y X Y) is a simple ordering in Y. The familiar ordering of the real numbers is a typical example of a simple ordering. In contrast, inclusion for sets is not, in general, a simple ordering. To point out the obvious, the applications of ordering relations are concerned with the determinations of orderings in various sets. In practice, ordering relations for a given set X are usually generated by assigned or proven structural features of X. That is, certain features of X, such as the existence of a particular type of operation or mapping property, will permit the definition of an ordering relation for X; an example of this nature appears in the exercises for this section. Properties of this ordering relation may then prove useful in deducing and describing further features of X. 'I'liercefore, it is convenient to have available terminology which gives primary emphasis to the set rather than to an ordering relation for it. A partially ordered set is an ordered pair (X, <) such that < partially orders X. A simply ordered set or chain is an ordered pair (X, <) such that < simply orders X. For example, if 9F is a collection of sets, then (9, C) is a partially ordered set. Again, if < is the usual ordering for the integers, then (Z, <) is a chain. From the standpoint of set theory, it is more economical to treat ordering relations than ordered sets, that is, sets with accompanying order relations. For example, if (X, <) is a partially ordered set, then < fl (X X X) is a partial ordering relation in X. Thus, instead of dealing with X and a relation < which partially orders it we can deal exclusively with the ordering relation < fl (X X X), since it determines X as its domain. 1.11 ( Ordering Relations 51 That is, all statements about ordered sets are equivalent to statements about their ordering relations, and vice versa. As an illustration of the preceding remark we restate our earlier characterization of < for a finite set X partially ordered by a relation <. if (X, <) is a finite partially ordered set, then x < y if there exists. < x = y in which each x, i., a chain of the form x = x, < xs < covers x,. This result enables one to represent any finite partially ordered set by a diagram. The elements of X are represented by dots arranged in accordance with the following rule. The dot
for x2 is placed above that for x, ifi x, < xz, and, if x2 is a cover of x,, the clots are joined by a line segment. Thus, x < y if there exists an ascending broken line connecting x with y. Some examples of such diagrams are shown below. o1 The first is the diagram of a chain with five members. Clearly, the diagram of any chain has this form. The last one is that of the power set of a set of three elements partially ordered by inclusion: the dot at the lowest level represents the empty subset, the dots at the next level represent the unit subsets, and so on. Such diagrams not only serve to represent given partially ordered sets by displaying the ordering relation, but, conversely, also may be used to define partially ordered sets; the ordering relation is just that indicated by the various broken lines. In preparation for our next definition in connection with partially ordered sets we discuss an example. The set 11, 2, 3, 5, 6, 10, 15, 301, whose members are the divisors of 30, is partially ordered by the relation < where x < y iff x is a multiple of y. It is left as an exercise to show that the diagram of this partially ordered set is identical to that given above for the subsets of a set of three elements partially ordered by inclusion. Although these two partially ordered sets are obviously not equal, they are indistinguishable so far as their structure as partially ordered sets is concerned. This is the essence of the identity of their respective diagrams. When this type: of relationship exists between two partially ordered sets it is certainly worthy of note, since any property 52 Sets and Relations! earn. 1 of one that is expressible in terms of its ordering relation has an analogue in the other. Thus, we propose to formalize this type of indiscernibility. The identity of the diagrams of the two partially ordered sets mentioned above implies, first, the existence of a pairing of the members of the two sets. 't'his can be formulated as the existence of a oneto-one correspondence, which has the advantage that it does not limit us to finite sets. Next, it is implied that the relationship between a pair of elements in one set, as specified by the ordering relation for that set, is the same as that for the. corresponding pair in the other set, relative to its ordering relation. The following definition is basic in the precise formulation of this property. A function f
: X -*- X' is orderpreserving (isotone) relative to an ordering < for X and an ordering <' for X' iff x < )' implies f(x) <' f(y). Then the likeness with which we are concerned can be described as the existence of a one-to-one correspondence such that it and its inverse arc order-preserving. The customary terminology in this connection follows. An isomorphism between the partially ordered sets (X, <) and (X', <') is a one-to-one correspondence between X and X' such that both it and its inverse are order-preserving. If such a correspondence exists, then one partially ordered set is an isomorphic image of the other, or, more simply, the two partially ordered sets arc isomorphic. Thus, the likeness which we observed between the collection of subsets of a three-clement set and the set of divisors of 30, with their respective partial orderings, may be expressed by saying that they are isomorphic partially ordered sets. When the concept of a partially ordered set was defined it was stated that a collection of sets partially ordered by inclusion is a typical example. This was rather loose talk, since the word "typical" has so many shades of meaning. One precise (and demanding) meaning that might be given is this: Each partially ordered set is isomorphic to a collection of sets partially ordered by inclusion. This is proved next. THEOREM 1 1.1. A partially ordered set (X, <) is isomorphic to a collection of sets, indeed, a collection of subsets of X, partially ordered by inclusion. Proof. For a in X define S. to be (x E Xjx < a]. 'l'inen the mapping f ore X into {Saga C X) where f(a) = Sa verities the assertion. The details are left as an exercise. This result is often stated as: "Each partially ordered set can be represented by a collection of sets (partially ordered by inclusion)." 1.11 1 Orde> ing Relations 53 In effect, the theorem means that the study of partially ordered sets is no more general than that of a collection of sets partially ordered by inclusion. In practice the transfer to such a partially ordered set is usually not carried out, since many individual partially ordered sets would lose much of their intuitive content as a result. Finally, we point out that the theorem does not assert
that each partially ordered set is isomorphic to a collection consisting of all subsets of some set. Such those of the form (4'(A), q), do not partially ordered sets, that is, typify partially ordered sets in general, since they have special features. For example, each contains an clement (namely, 0) less than every other element and an element (namely, A) greater than every other clcrncnt. We conclude this section with the introduction of further terminology for partially ordered sets that will be employed later. A least >nember of a set X relative to a partial ordering < is a y in X such that y < x for all x in X. If it exists, such an element is unique, so one should speak of the least member of X. A minimal number of a set X relative to < is a y in X such that for no x in X is x < y A minimal member need not be unique, as the second diagram above illustrates. A greatest member of X relative to < is a y in X such that x < y for all x in X. A greatest element, if it exists, is unique, so one should speak of the greatest element of X. A maximal member of X relative to < is a y in X, such that, for no x in X is x > y. A partially ordered set (X, <) is well-ordered iff each noncrnpty subset has a least member. A familiar example of a well-ordered set is the set of nonncgative integers relative to its natural ordering. Any wellordered set (X, <) is a chain, since for two distinct elements x and y of X the set {x, y) must have a first. element, and hence either x < y ory <x. If (X, <) is a partially ordered set and A C X, then an clement x in X is an upper bound for A iff, for all a in A, a < x. Similarly, an element x in X is a lower bound for A if, for all a in A, x < a. A set may have many upper bounds. An element x in X is a least upper bound or suprcrnum for A (symbolized, tub A or sup A) i(f x is an upper bound for A and x < y for all upper bounds for A. In other words, a suprcrnum is
an upper bound which is a lower bound for the set of all upper bounds. An element x in X is a greatest lower bound or infinum for A (syrnboli2ed, glb A or inf A) iff x is a lower bound for A and x > y for any lower bound y for A. It is immediate that if A has a least upper bound, then it is unique, and that the same is true for a greatest lower bound. 54 EXERCISES Sets and Relations I c 11 A t'. 1 11.1. Show that if p is a partial ordering relation, then so is 11.2. For the set of real-valued continuous functions with the nonnegative reals as domain, define f = O(g) to mean that there exist positive constants M and N such that f(x) < Mg(.%.) for all x > N. Show that this is a preordering, and define the associated equivalence relation. 11.3. If < is a partial ordering in X, show that < is an irreflexive and transitive relation in X. Conversely, if < is an irreflexive and transitive relation in X, show that the relation < such that x < y iflf x < y or x = y is a partial ordering in X. 11.4. For what sets A is ((P(A), C) a simply ordered set? 11.5. Let (X, <) and (X', <') be partially ordered sets. Show that X X X' is partially ordered by p where (x, x')p(y, y') iff x < y and x' <'y'. The partially ordered set (X X X', p) is the (cartesian) product of the given partially ordered sets. 11.6. The dual of a partially ordered set (X, p) is the partially ordered set (X, p) (see Exercise 11.1). If (A, <) is a partially ordered set and a, b C X with a < b, then the set of all x in X, such that a < x < b, is called the closed interval [a, b l. Show that the set of intervals of a partially ordered set (.V, <), partially ordered by inclusion, is isomorphic to a subset of the product of (X, <) and its dual. 11.7. A partially ordered set is self
-dual if it is isomorphic to its dual. Show that (a) there are just two nonisornorphic partially ordered sets of two elements, both of which are self-dual, and (b) there are five nonisornorphic partially ordered sets of three elements, three of which are self-dual. 11.8. Show by an example that if (A', <) and (X', <') are partially ordered sets and f : A'-->- X' is a one-to-one correspondence which preserves order, then f -r need not preserve order. 11.9. Given that f is an isomorphism between the partially ordered sets (X, <) and (X', <'), show that x < y ifl' f(x) <'1(y). 11.10. Supply details for the proof of Theorem 11.1. 11.11. Let (X, <) be a partially ordered set. Show that a is a maximal element iffy C X and y > u imply y = u. Show that v is a minimal element ill yEXand y <viniplyy = v. 11.12. Let 5; be the collection of all subsets of Z+ which have at most n members for n a fixed positive integer, and let t be the collection of all finite subsets of V. Show that, relative to inclusion, (a) each element of IT. having n members is maximal, and (b) fl has no maximal elements. 11.13. As the elements of a set X we take all square regions which lie inside Bibliographical Notes 55 a given rectangular region which is itself not a square. Relative to inclusion, what are the maximal elements of X? 11.14. Show that in a chain the notions of a greatest element and a maximal element coincide, and show the same for a least element and a minimal element. 11.15. Let (X, <) be a partially ordered set with the property that each nonempty subset which has an upper bound has a least upper bound. Show that each nonempty subset of X which has a lower bound has a greatest lower bound. 11.16. Show that if (X, <) is a well-ordered set, then it has the property assumed for the partially ordered set in the preceding exercise. 11.17. Let X be a set and p an operation in X. (Thus, p is a function on
.1, X X into X; let us denote the value of p at (x, y) by xy.) Suppose that p is cornmutative, associative, and idernpotent ]that is, xy = yx, x(yz) = (xy)z, and xx=xforallx,y,zCA].Forx,yEXdefinex<yiflx=xy.Show that (a) < partially orders X, (b) if X has a least element 0, then Ox = 0, (c) xy < x, y and, ifz < x, y, then xy > z. 11.18. The relation < where in < it ifTin divides it partially orders Z'. Show that each pair of integers has a least upper bound and a greatest lower bound relative to this ordering. 11.19. Show that each subset of (1'(A) partially ordered by inclusion has a least tipper bound and a greatest lower bound. BIBLIOGRAPHICAL NOTES Sections 1-2. For a detailed historical survey of Cantor's work see the intro- duction by,)ordain in Cantor (1915). Sections 3-10. An excellent account of elementary set theory appears in Ilamilton and Landin (1961). Section 11. A high-level account of the theory of partially ordered sets may be found in BirkhofT (1948). CIIAPTLK 2 The Natural Number Sequence and Its Genera lizabons T t r s c: II with a formulation of a precise definition of the natural number sequence 0,1,2,... (where we rely on the dots " " to suggest the. continuation of the sequence beyond the numbers displayed) from an intuitive description of this set. This definition is taken as the basis for the definition of two operations in this set. The result is the system consisting of the natural numbers, the operations of addition and multiplication, and the familiar ordering relation-all of which the reader has known since childhood. Although in certain respects Section 1 acids nothing to his knowledge, it should be of interest to him to find how few assumptions are required to derive the familiar properties of the natural numbers. Section 2 discusses definition and proof by induction. Section 3--Section 7, and Section 9 give an account of Cantor's trans(initc aritluuctic. 'this consists of the continuation of the natural number sequence, first with respect
we call the successor function and symbolize it by '. In ternis of this function the remainder of our description can be expressed in two properties. Ni. N2.'is a one-to-one mapping on N into N - (0}. If M is a subset of N, such that 0 C M and m' C M whenever m C M, then h1 = N. Property N2 (which has its origin in the assertion that the succession of discrete steps -consisting of starting with 0 and repeatedly passing from a number to its successor-yields all of the natural numbers) is 58 The Natural Number Sequence and its Generalizations I C u A Y. 2 the basis for the principle of induction. Writing P(n) for "the natural number n has property P," we state this as follows. If P(O) and, for each natural number in, P(m) implies P(m'), then P(n) for each natural number n. 7'he proof follows immediately from N2 upon consideration of (rn C NI1'(rn)}. In order to arrive at a precise formulation of the natural number system, starting from the foregoing description, it is convenient to make the following definitions. A triple (X, g, xo), where X is a set, g is a unary operation in X (that is, a function on X into X), and xu is an element of X, is a unary system. An integral system is a unary system (X, s, x(,) such that 11. I1. s is a one-to-one mapping on X into X if Y is a subset of X such that xu E Y and ys C Y whenever and yE_ I',then Y=X. Thus, our description of the natural number system may be surrt- rnarized by the assertion that (N, ', 0) is an integral system. Before giving other examples of integral systems we call attention to one consequence of I, and 12- that s is a mapping onto X - {xu}. This follows from the fact that (xu} U }X f s = X, which is a consequence of 12. EXAMPLES 1.1. In spite of the self-imposed restrictions stated in the introduction to this chapter, we are free to use the real number system for illustrative purposes. Thus we can introduce the following further examples of integral systems. (a)
The numbers a, a 1- d, a -I- 2d, of an arithmetic progression (in which a, d are real numbers with d ;t 0), the map s of this set into itself with xs = x 1- d, and the number a. (b) The members a, or, ar2, of a geometric progression (in which a, r are real numbers with a 0 0 and 0 < r 0 1), the function s mapping x onto xr, and the number a. 1.2. As a preliminary to the observation that by virtue of our initial description-the natural number sequence qualifies as an integral system, we might have mentioned that it has the following properties. 1'1. 0 is a natural number. P2. If n is a natural number, then n' is a uniquely determined natural number. P3. For all natural numbers in and n, if rn' = n', then in = n. 2.1 I The Nulural Number Sequence 59 p,. For each natural number n, n' 0 0. P. If M is a subset of N such that 0 C M and m' C M whenever in E iLI, then M = N. These are the now famous Peano axioms for the natural number system. In a book published in 1889, G. Peano took these as a point of departure for an axiomatic development of the natural number systems. The axioms themselves are actually due to R. Dedckind (1888). It is worthy of mention that each P;, _ = 2, 3, 4, 5 lists exactly one property of (N, ', 0) in addition to those appearing i-,. Properties P,-P4 are simply a breakdown of Nr into "atomic" inin Pigredients while P5 is N2. Conversely, starting with an integral system (X, s, properties that imitate P1-P5 may be asserted. Our immediate goal is to prove that any two integral systems, (X, s, xo) and (Y, t, yo), are isomorphic; that is, there exists a one-to-one correspondence f between X and Y with f(xo) = yo and f(xs) _ (Jx)t for all x in X. This means that the elements of X can be paired with those of Y in such a way that successors of corresponding elements correspond. For the proof a definition is
required. Let (X, g, xo) be a unary system. The set of descendants of xo under g (in symbols, Dvxo) is the intersection of all subsets A of X, such that xo C A and xg C A whenever x E A. (This latter requirement will often be phrased as "A is closed under g.") Such subsets A exist; indeed, X is one. Two characterizations of a set of descendants are given next. LEMMA 1.1. Let (X, g, xo) be a unary system. Then Dvxo is the smallest subset of X which contains xo and which is closed under g. Alternatively, x C Ddxo iff x = xo or there exists a y in Dvxo such that x = yg. Proof. The proof of the first statement is left as an exercise. For the second, consider an element x in xo. Suppose x 0 xo and that there does not exist y in Dgxo such that x = yg. Then Dvxo - }x} is a proper subset of which contains xo and which is closed tinder g. This is a contradiction of the first statement in the lemma. Either x = xo or x LEMMA 1.2. Let (X, s,.ro) be an integral system and (Y, t, yo) be a unary system. Define sVi: X X X Y with (x, y)sVt = (xs, yl) Then (X X Y, s Vi, (xo, yo)) is a unary system. Iff is the set of descendants of (xo, yo) under s Vt, then 60 77,e Natural Number Sequence and its Generalizalions I c.II A P. 2 (1) f is a function on X into Y, (II) fxo = yo and f(.rs) = (fx)t for all x in X, and (1II) f is uniquely determined by the properties in (II). REMARK. To assist with the understanding of (I), we suggest the study of an example such as the following. Let the integral systern be (N, ', 0) and the unary system he ({a, b, c, d), t, c) where t = {(a, a),
(b, c), (c, d), (d, a){. Let us determine the set of descendants of (0, c) under the!unction formed froth' and I by the rule given. Along with (0, c) this set must contain (1, d) and hence, (2, a). Since at = a, the only further members present are those of the form (n, a). Clearly, this set is a function on N into {a, b, c, di. for n = 3, 4, Proof. That (X X I', s Vt, (xo, yo)) is a unary system is clear. By definition, f is the intersection of the collection a of all sets A, such that A 9 X X Y, (x0, yo) C A, and (x, y) C A implies (xs, yl) C A. That f is a function on X into Y is the fifth property off appearing below. The first four are left to the reader to verify. (1) f C a. (2) fCAforeachAin(t. (3) f is a relation with X as domain. (4) u C f iff u = (xo, yn) or there exists (x, y) C f with u = (Xs' yt). (5) f is a function on X into Y To establish (5) we prove by induction (that is, using the property 12 of the integral system (X, s, xo)) that for all x in X, (x, y) and (x, z) in f imply y = z. Let Z consist of all elements of X for which this is true. 't'hen xo C Z. Indeed, suppose that along with (x(,, yo), which. is in f by (4), also (x0, y,) C f, where y, /`- yn. By (4), (x0, )',) = (xs, yt) and hence xo = xs, which is impossible. Hence, the basis for the induction follows. Assume next that x C Z. If (xs, y,) and (XS, Y2) are in f, then by (4) and the assumption that s is one-to-one, there exist y3, y4 C 1', such that (
x, y3) and (x, y4) are in f. From the induction hypothesis it follows that y3 = Y4 and hence yj = y2. This completes the proof of the induction step. Ilence Z = X and the proof is complete. For (I1) there remains to prove that f(xs) = (fx)t for all x in X. If x C X, then for exactly one y in F, (x, y) C f and, further, (xs, yt.) C f Writing "fx" for "y and "f(x.s)" for "yt," we!have f(xs) = yl = (fx)l. For (111), let g: X--*- Y, such that gxn = yo and g(xs) _ (gx)i for all x in X. Let Z be the set of all x in X, such that fx = gx. 't'hen xp C Z. Assume that x C Z. Then f(xs) _ (fx)t = (gx)t = g(xs). Hence, xs C Z and hence, Z = X. That is f = g. 2.1 I The Natural Number Sequence 61 T1IEOREM 1.1. Any two integral systems are isomorphic. Proof. Let (X, s, xo) and (Y, 1, yo) be integral systems. According to Lemma 1.2 there exists a function f: X--->- Y, such that fro - y and xo and g(p) = f(xs) = (fx)t and a function g: Y - Xsuch that,gyo (gy)s. We contend that g o f = ix, the identity function on X. Let Z he the set of all x in X such that (g f )x = x. Clearly, x C Z. Further, if x E Z, then (g °1)(xs) = g(1(xs)) = g((fr)l) = (g(fx))s = ((g ° f)x)s xs. That is, if x E Z, then xs C Z. Hence 7, = X and g -f - i x. Similarly, f o.g = iy. Together, these results imply that f is a one-to-one correspondence between
X and Y. Finally, fxo = yo and f (xs) = (fx)l, so the systems are isomorphic. This is a significant result for us. To insure that it is understood, let us review the pertinent facts. Our initial (and purely intuitive) description of the natural number sequence led us to conclude that it is an integral system. Such an observation in itself gives no indication of the degree to which it captures those features and only those features which we intuitively assign to the natural number sequence. I hcorcun 1.1 gives us precise (and satisfying) information on this score, for it asserts, in effect., that apart from notation used there is only one integral system. Thus, the statement that the natural number sequence is all integral systcrn amounts to a complete description. This we lake as our formal de/inition of the natural number sequence. What this conies to is fixing on one particular integral system and designating its initial clement by 0, its successor by 0', and so on. To expedite our development of properties of the integral system (N, ', 0) we derive another consequence of Lemma 1.2. 'I'll E 0 It EM 1.2. Let B be a noncrnpty set, c be an clement of B, and g be a function on N X B into B. Then there exists exactly one function k: N B such that k(0) = c and k(n') = g(n, k(n)). Proof. Define 1: N X I3 -* N X 13 where (n, b)t = (n', g(n, b)). Applying Lemma 1.2 to the integral system (N, ', 0) and the unary system (N X 13, t, (0, c)) yields a function f : N -* N X B where f0 = (0, c) and fn' = (fn)l. 62 Die Natural Nuinber Sequence and its Generalizations i CHAP. 2 We assert that fn' = (n', g(fn)) for n C N. Since JO' = (fO)t = (0, c)t = (0', g(0, c)), n = 0. Assume that it is true for n and consider it for n'. We have the assertion is true for fit" = (Jn')t = (n', g(fn))
t = (n", g(it', g(fn))) _ (n", g(fn')), as required. Now define k: N - B where k(O) = c and k(n') _ (g o f)(n). Then k(O') (g -f)(0) = g(f0) = g(O, c) = g(O, k(0)) and k(n") _ g(fn') = g(n', g(fu)) = g(n', k(n')). Hence, k(0) = c and k(a) _ g(n, k(n)) for n C N. 't'hat k is unique is shown by a straightforward induction proof, which is left as an exercise. We turn now to the definition of an ordering relation for N. The basis for the intuitive ordering of the natural numbers is the order in which they are generated. One says that in is less than n iff m is generated before is in the course of generating n or, what amounts to the same, m is less. This hhra;in; than or equal to n if n = in or n = m' or it = in" or is the origin of our definition of < for the integral system (N,', O. For in, n in N, we define m<n if n C Dm, the set of descendants of rn under '. Those properties of sets of the type Dm which will prove useful in developing consequences of this definition are listed next. D,. Dn = }n} U Dn'. 1)2. Dn' 9 [Dn]', the set of successors of elements of Dn. 1)3. D,. Dm = Dn implies that in = it. 1)2. If 0 C M C N and Al is closed under ', then Al = Dk for a n V Dn'. uniquely determined k C N. Proofs of 1), and 1)2 are left as exercises. We prove 1)3 by induction. It is true for n = 0, since the contrary (0 C DO') implies that 0 is a successor. Assume that n (Z Dn'; to prove that n' (7 Dn". Assume, to the contrary, that n' C Dn". Since Dn" c JDn' j' by 1)2j we have n' = q' for
4. For the integral system (N, ', 0) there exists ex- actly one function a: N X N -} N such that (1) for each n in N, a(0, n) = n, and (I1) for all m and n in N, a(m', n) = (a(in, n))'. This function is addition in N; a(m, n) will be abbreviated to m + n. Proof. Let n be a fixed element of N. Define g: N X N ->- N where g(x, y) = y'. According to Theorem 1.2 there exists exactly one function a,,: N -'- N where a (0) = n and g(m, for in, n in N. Clearly this function satisfies (I) and (11). To prove its uniqueness, let -y be any function on a by a(m, n) = 64 The Natural Number Sequenre and its Generalizations I CI I All. 2 N X N into N such that y(0, n) = n and y(m', n) = (y(m, n))'. For each n define yn: N --} N by Then y(0, n) = it and 'yn(m) = y(m, n). -yn(m') = y(nt', 71) = ('y(m, n))' = (yn(rn))'. It follows from Theorem 1.2 that y = a,, for each n. Thus, for in, n in N, a(m, n) = an(m) = yn(m) = y(m, n). 1-lence, a = y. TI-IEOREM 1.5. Addition in N has the following properties. A1. Associativity. For all in, it, and p in N, in + (n -}- p) = (in 4- n) -1- p. A2. Cornmutativity. For all in and n in N, in -1- n = n + in. A3. Cancellation laws. For all in, n, and /p in N, p + in = p + n implies m = n and in -I- p = n -I- /i implies in = n. A4. For all in and n in N, in <
it if there exists p in N such that p -1- in = n. A,. For all m,n,and pin N,m<niflp+in <pA-n. A6. For all in and n in N, in -- a = 0inipliesin = 0 and n = 0. I'ioof. We verify in turn Al---AB. In the notation adopted for addition, its assigned properties appear as 0 -1- n = it and in' -1- n = (in -F n)'. A1. Let n and p be fixed and let 141 = [nl C NIn: + (n + p) = (m -F n) -F p}. Then 0 C M since 0 A- (n -I- p) m C M. 7'licn (0 -I- n) -1- p. Assume that in' -1 (n - p) = [m -I- (n -1 p) (m 4. n) + p l' = (in + n)' -I- p = (in' 4 n) -I p so that m' C M. IIence M = N and the proof is complete. A2. As a preliminary step we prove that for a fixed in, n' + in = n -I' in' for all n in N. 't'his is true of 0, since 0' + in = (0 -1- m)' = in' - 0 -1 in'. Further, if it is true of it then it is true of n' since (n')' -4- in - (n' + in)' = (n -I- in')' _- n' -1- m'. 'I'lu assertion then follows by 12. Iris applied in the last step of the proof of the next statement.. 11' for it fixed, N _ N. Indeed, jim C N, n4-m}, then N 2.1 Die Natural Nuinbei Sequence 65 then in' -f- n = (in + n)' _ (n + m)' = n' -I- m = n -1- na'. That is, an' C N. We can now prove A2. Clearly 0 C No, and this with the inclusion N;, C N implies that No = N. Hence 0 C 1V,,, and this with N,
', C N implies that N = N, which proves A2. A3. We prove the contrapositive of the first statement. If in, n are distinct fixed natural numbers, then p -f m 5,6 p + n for all p in N. Clearly 0 C Ip C Nip + in Fl- P + n 1. Assume that p is a. member of this set. Then p + in 0 /i + n, from which it follows that (p -I- in)' x (p + n)', or/i' A- in :X p' --I- n. This completes the proof that /i + in = p -I- n implies in = n. The second assertion then follows, using A2. l.ct in and it be natural numbers with in < It. Then A4. Ix C Nix ± in n) is noneanpty (indeed, one can prove by induction that n -1- in > n for all in and n) and consequently has a least member p by Theorem 1.3. Either p -1- m = n or p + in > n. Assume that p -1- in > n. Then clearly p 0 0 and hence p is the successor of a natural number q. 't'hus q' + in = (q -1-?it)' > n, which implies that q + in > n. Since q < q' _ /i, this yields a contradiction. Ilenee /5 -I- in = n. The converse, which asserts that if p + in = a then in < n, follows from the relation in < p -I- in mentioned above. If an < n, then d -I- in = n for sonic d 0 0, using A4. Ilence p-I-n=p+(d+in) _ (p +d)+in = (d -1 p) A-in = A5. Thus, by A4, P -I- in < p -I- as and, since d 0 0, the strict inequality p + in < p + n follows. The proof of the converse is left as an exercise. A6. We shall prove the contrapositivc statement. If in,-*`- 0 or 0, then in -I- n - 0. Assume that in ; 0. Then there n exists p such that in
= p'. I fence in + it = (p -I- n)' and, consequently, in + as 0, being a successor. Similarly, if n 04 0, 0. Thus, if in or n is different from zero, sP then in -I-- as is in + it. Multiplication of natural numbers, as understood intuitively, enjoys the following two properties, among others. For all natural numbers in 66 7'he Natural Number Sequence and its Generalizations I e i i A P. 2 and n, On = 0 and m'n = mn -I- n. According to the next theorem it is possible to define exactly one operation µ in N, regarding (N, ', 0) as an integral system, with these two properties. Accordingly, µ takes on the role of the only possible candidate for an operation in N which might the properties of intuitive multiplication. That it does is have all anticipated by our calling y "multiplication" from the outset and designating the value of it at (in, n) by inn. T H F O R I? M 1.6. For the integral system (N, ', 0) there is exactly one function µ: N X N -* N such that (I) for each n in N, µ(0, n) = 0, and (II) for all m and n in N, 4(in', n) = p (m, n) + n. This function is multiplication in N; µ(m, n) will be abbreviated to m n or simply Mn. Proof. Let n be a fixed element of N and let g: N X N -r~ N where g(x, y) = y + n. According to 'Theorem 1.2 there exists exactly one function µ,,: N --} N where y. (O) = 0 and u,,(m') = g(rn, U,,(rn)) = n. for in, n in N. Clearly this function Now define µ by µ(m, n) = satisfies (I) and (11). Its uniqueness may be inferred from that of µ for each n. Ti I EO R F.M 1.7. Multiplication in N has the following properties. Mr. Associativity. For all in, it, and p in N, m(np) = (mn)p. M2. Comrnutativity. For all in and n in N,
nut = nin. M. Cancellation laws. For all in, n, and p in N, p 0 and pin=pn orm/i =u imply 7n = n. M9. I)is(rihutivity over addition. For all in, n, and p in N, rn(n + p) = inn -{- nip and (n + pi)in - nm + pin. M6. For all in, n, and p in N, /' 34 0 implies that m < n ifI' PM < P11. M6. For all in and n in N, mn = 0 implies that in = 0 or n = 0 or, what is equivalent, if in 0 and n r 0, then mn 0 0. It is convenient to prove these properties in the order Ma, Mx, hoof. M 1, M6, M3, M. 2.1 I I he Natural Number.Sequence 67 M4. For fixed n and p, consider {m C NIm(n -I- p) = inn -I- rnp}. Clearly 0 is a member of this set, and if m is a member then so is m', since M2. m'(n + p) = m(n 4- p) + (n + p) _ (mn + mp) -I- (n + p) _ (nnn + n) + (nip + p) = m'n + m'p. This establishes the first distributive law in M4. The second follows from this and M2, which we prove next. It is left as an exercise to show by induction that for all n in N, ri0 = 0 and nO' = n. Assuming these preliminaries, fix n and consider {rn C Njmn = nm}. This set contains 0 since On = 0 = nO. Assume it contains in. Then it contains m', since m'n = inn+n = nnz+n = nm + nO' = n(m + 0') = n(0' + m) = nm', where we have used the preliminary result n'O = n and the one distributive law already proved. Hence M2 follows by the principle of induction. M,. For fixed n and p consider {m C NIm(np) = (mn)p}. This set contains 0,
and if it contains m then it contains in', since m'(np) = m(np) + np = (mn)p + np = (mn + n)p = (m'n)p. M6. Assume that mn = 0 and in 0 0. Then in = p' for some p. Hence 0 = inn = p'n = fin -I- n and n = 0 by A6. M3. Assume in = pn and p 0. Since < simply orders N, either rn < nor n < in. If rn < n, then by A4, n = d + in for some d. Then 0 -}- pm = pnz = pn = p(d -I- m) = pd -I- pm. Hence, by A3, 0 = pd. 't'his and p 0 0 imply that d = 0, by M6. hence in = n. The proof is similar starting with n < in. The other cancellation law then follows, using M2. M5. Assume that m < n. Then, by A3, n = d + in for some d F& 0. Hence pn = pd -I- pm where pd 0 0 by M6. I lence pm < din by A4. The converse is left as an exercise. We have stressed the fact that the preceding definitions of an order relation and the operations of addition and multiplication in N are based solely on the assumption that (N, ', 0) is an integral system. It is in order to prove that the indiscernibility of two integral systems, as 68 The Natural Arrnnlcr Sequence and its Generalizations, CIIAP. 2 described in Theorem 1.1, extends to the case where the ordering relation, addition and multiplication are incorporated into each. TiIEOR EM 1.8. Let (X, s, xu) and (X, s, x;) be integral systems. Let -+-,, and < be the addition, the multiplication, and the ordering relation, respectively, in X which satisfy the earlier definitions. Let , and < be the corresponding relations in X. Then there + , exists a one-to-one mappulg f on X onto X* which preserves each of these relations in the following sense: (I) Ax + y) = f(x) +*f(
y), f(y), (2) f(x y) = f(.Y) (3) x < y iff f(x) <* f(y) 1.1 there exists a one-to-one mapping Proof. According to f on Xonto X* such that f(xo) _ xi, and f(xs) = (f(.x))s. This rilapping fulfills the conditions (1)-- (3). To prove that (1) holds we fix y 't'hen xo E Y, and consider Y =I x C X l f (x +y) ---.f(x) -h- f (y) }. f-* f(y). Also, if x C Y, since f(xo + y) = f(y) = xU +* f(y) = f (xo) then f(x.s -H- y) = f((x + y)S) = (f (X + y))S* = (AX) + * f(y))s* = (f(x))S* +f(y) f(xs) +f(y), so XS C 1'. Hence Y = X. The proof that (2) holds for f is left as an exercise. That f preserves the ordering relation in both directions may be inferred from (1) and the characterization of the ordering relation in terms of addition given in A4 of "Theorem 1.5. This concludes the first. stage of the derivation of basic properties of the natural number sequence regarded as an integral system. Upon abbreviating 0' by 1, the successor n' of n can be written as n. 4- 1, since n' = (0 f- n)' = it + 0' = n + 1, and we shall henceforth do so. In the next section deeper properties concerning definition and proof by induction are considered. Among the applications discussed is the unique factorization theorem for N. As a consequence of Theorem 2.2 there follows the general associative laws for addition and multiplication, which generalize A, and Mi. Among the exercises for Section 2 appears the general commutative law For any commutative composition; this yields commutative laws for addition and multiplication which gen- I The Natural Number Sequence 2.1 69 eralize A2 and M2. Finally, a general distribut
ive law can be derived from M4 by induction. In brief, regarding the natural number sequence as an integral system, all of the familiar arithmetic of the natural numbers can be derived. EXERCISES I.I. Show that a set X, together with a function f, determines an integral system provided that (i) f is a one-to-one map on X onto a proper subset of X,, and (ii) whenever Y is a subset of X such that Y contains an element of X [XJ f and [Y]f S Y, then Y = X. 1.2. This exercise is concerned with the Peano axioms in Example 1.2. So they may be considered objectively, we rewrite them as assumptions about a set X. P. xo C X ; that is, X is nonempty.'is a mapping on X into X. P*. Pa. If x, y C X, and x' -y', then x = y. I'4 IfxCX, thenx',-ti`X1). P;. If 1' C A' and xQ C Y and, whenever y C Y then y' C Y, then Y - X. Show that P;-P4 imply that X, together with the function defined in P2', and xo form a unary system which satisfies I. 1.3. Construct examples of systems which satisfy each combination of four of the five Peano axioms in Exercise 1.2 but violate the remaining one. 1.4. Complete the proof of Lemma 1.1. 1.5. Complete the proof of Lemma 1.2. 1.6. Complete the proof of Theorem 1.2. 1.7. Establish D, and D2 as properties of descendants. 1.8. Establish D5 as a property of descendants by first proving that if n tZ M, then M C Dn'. Deduce that if, in addition, n' C Al, then Al = Dn'. 't'hen proceed with the proof of D6 by considering the case where 0 C M and that where 0 (7 M. 1.9. Let X be a set, g: X -'- X, and n be a fixed element of N. Show that Lemma 1.2 implies that for each x in X there exists exactly one element y in X such that (n, y) is a member of the set
of descendants of (0, x) under 'Vg. The resulting function on X into X we designate by g". Show that g" = ix, g' = g, and g"' = g" o g for all a in N. 1.10. Let /3: N X N --} N with 13(in, n) = ns'" where s is the successor func- tion on N and s" is defined in Exercise 1.9. Show that 13 is addition in N. 1.11. Complete the proof of A4 and A5 in Theorem 1.5. 1.12. For n in N define t,,: N ->- N by at,, = a I n. Show that the function v: N X N - hl with v(m, n) = Ot. is multiplication in N. 1.13. Complete the proof of M5 in 'I'hcorcm 1.7 1.14. Theorem 1.4 is applicable to any integral system. Determine the func- 70 The Natural Number Sequence and its Generalizations I CI I A r. 2 tion a of Theorem 1.4 for each of the integral systems (a) and (b) defined it, Example 1.1. 1.15. Theorem 1.6 is applicable to any integral system. Determine the func- tion µ of Theorem 1.6 for the same two integral systems. 1.16. Using Theorem 1.5, show that in N (a) if x + u = y and y + v = x', then either u = 0 or v = 0. 1.17. Prove that for a, b in N with b 7 0, there exist unique elements q and r of N, such that a = qb + r where r < b. This is the division algorithm for N. 1.18. Let S be a set such that there exists a one-to-one mapping F on S onto a proper subset of S. 'I'hen F induces a mapping f on 6'(S) into 6'(S) in an obvious fashion and (61(S), f, S) is a unary system. Form 1))S and define s: DfS -*- DfS by As = f(A). Show that (DfS, s, S) is an integral system. 2. Proof and Definition by Induction In the preceding section
we described and repeatedly used the principle of induction as a method of proof. 't'here is a second form of this principle which also finds many applications. To distinguish the two, let us call that one already discussed the principle of weak induction. In weak induction, to prove that P(n) for all natural numbers n, one proves P(0) arid then derives P(m + 1) from the assumption that P(m). In the second form of the principle, which we call the principle of strong induction, one assumes each of P(0), P(1),, P(m) and uses them to derive I'(m + 1). With more assumptions, in general, it is easier to derive P(m + 1). Hence, strong induction finds applications as a method of proof where direct application of weak induction would be difficult. A precise formulation of the principle follows; as before, P(n) stands for "the natural number n has property P." If P(0), and if, for each natural number m, P(r) for all r < in implies P(m + 1), then P(n) for each natural number n. The validity is an immediate consequence of the following theorem and is left as an exercise. THEOREM 2.1. Let M be a set of natural numbers such that (1) 0 C M, and (II) if rCMfor each r <rn,then m+I C M. ThenM=N. 2.2! Proof and Definition by Induction 71 Proof. Consider N - Al. If this set is nonempty then it contains a least member, byTheorern 1.3. This number is not 0 by (1) and hence may be written in the form m ± 1. Then, for each r < m, r E Al. By (11) it follows that m -I- 1 E Al, contrary to the choice of '?n 4- 1. Thus, the assumption that N - Al is nonempty leads to a contradiction. Ilence Al = N. Our formulation of both the principle of weak induction and that of strong induction has been for the case where the induction begins with 0. Each case can be generalized to start with any natural number no. In this circumstance the conclusion reads "for all natural numbers n > no, EXAMPLES 2.1. As an illustration of a proof by strong induction we prove the theorem that every integer greater than I
has a prime factor, starting the induction with 2. Obviously 2 has a prime factor. Assume the theorem for all m with 2 < m < n and consider n -1- 1. If n -- 1 has no factor a with 1 < a < n -I- 1, then n + 1 is a prime and has itself as a prime factor. If n + 1 has a factor a with I < a < n -1- 1, then 2 < a < n. By the induction hypothesis a has a prime factor b, which is then a prime factor of n -I- 1. 't'hus, in every case, n -I- I has a prime factor. 2.2. As a somewhat more important illustration of proof by strong induction, we prove next what is often called the fundamental theorem of arithmetic: Every natural number greater than 1 has a representation as a product of primes that is unique to within the order of the factors. Again we begin the induction with 2. Clearly, 2 has such a representation. Assume that all numbers less than n have unique representations and consider n. The set of divisors of n which are greater than 1 is nonempty and, hence, has a least member p. Then p is a prime since a divisor q of p with 1 < q < p would be a smaller divisor of n. If n = pram, then n, has a unique representation by the induction hypothesis. Replacing nm by its unique representation as it product of priunes yields a representation of n = pnm as a product of primes, and this is the only representation of n which contains p as a factor. If the theorem is false for n then it has a second representation. If q is the smallest prime present in this second representation, then q > p, since this other representation of n does not involve p and p is the smallest divisor (> 1) of n. Let n = qn2 and q = p + d. Then n = pn2 -I- dn2. Since p divides n, p divides dn2. Now dn2 < n and, consequently, has a unique representation. I lence, p divides d or p divides n2. But p is not a factor of n2 since it contains no factor less than q and q > p. Thus p divides d. Let d = rp. Then q = P + rp = p(1 + r
). This is a contradiction 72 The Natural Number Sequence and its Generalizations I C If A P. 2 since q is a prime. Thus n has no decomposition other than the essentially unique decomposition with p as a factor. We consider next definition by induction. Two examples which have already been given are that of addition and that of multiplication in N. These were justified, we recall, by an appeal to 'T'heorem 1.2. A definition which can be justified by an appeal to Theorem 1.2 is called a definition by weak induction. Other examples of this type of definition are that of b" (for a real number b and a natural number n) as b°=1 and n! (for a natural number n) as 0! = 1, (n - { - 1)! (n + 1) n!. The reader may question the necessity of resorting in such cases to Theorem 1.2 or, what amounts to the same, the complexity of the proof of Theorem 1.2. For he may be satisfied with the following argument that the two conditions k(0) = c, k(n -{- 1) = y(n, k(n)), where-- restating the hypothesis of Theorem 1.2, c is it given constant, and g is a specified function of two arguments- do define it function k. Clearly (so the argument goes) the two conditions define k(0). Then with the choice of 0 for n in the second, k(1) is specified: Next, setting a = 1 in the second condition, k.(2) is specified: k(1) _ g'(0, k(0)) = g(0, c). k(2) = g(1, k(1)) = g(1, g(0, c)). Proceeding in this manner, k(n) is uniquely specified for any given natural number and only such. Thus, a function whose domain is N has been defined. There is an error in this intuitive reasoning. To disclose it we recall that a function is a set, so that to define a function is to define a set (of a certain kind). The procedure just employed permits one to define as many members as he chooses (namely, (0, k(0)), (1, k(1)), -, (n, k(n)), for any preassigned ni of a certain set, but it does not yield
a definition of the set consisting of all such ordered pairs, unless the function which the intended set is to delinc is already known. In brief, the error consists in using a function symbol without first giving a function for it to denote. 2.2 1 Proof and Definition by Induction 73 Adrniuedly, the intuitive argument does make it plausible that the two conditions define exactly one function, and the proof of "I'hcorcnn 1.2 settles the matter. Another instance of a definition by weak induction (as well as a proof by weak induction and one by strong induction) occurs in the derivation of the general associative law for an arbitrary associative operation in a set. Reference has already been made to this result. The setting in which to view it may be described as follows. Up to this point we have considered several (binary) operations in various sets. By definition, these are functions of the form f : X2 -> X where X is some set and X2 is an abbreviation for X X X. Each of the following notations has been used for the image of Via, b) under f at one time or another: a U b, a fl b, a o b, ab, a -I- b. In order to achieve impartiality so far as notation is concerned in this discussion, we shall use a * b for the image of (a, b). In terms of f, two ternary operations in X that is, mappings on X3 into X- may be defined. One of these maps (a, b, c) onto (a * b) * c and the other maps (a, b, c) onto a * (b * c). Similarly, a total of five 4-ary operations in X may be defined in terms of f. These are the mappings on X1 into X, such that the image of (a, b, c, d) is one of ((a * b) * c) * d, (a * (b * c)) * d, (a * b) * (c * d), a * ((b * c) * d), a * (b * (c * d)). In like fashion f serves to generate n-ary operations in X for n > 4. For an arbitrary n (> 2) let us call the image in X of (a,, a2,, a") in X" under an n-ary operation originating with f a composite
of a,, a2,, an (in that order). Such an entity is simply the string a, * a2 * * a", together with sufficient parentheses to specify unambiguously n - 1 applications off. If f has the property that, for all a, b, and c in X, a(bc) = (ab)c, that is, f is an associative operation in X (or satisfies the associative law,, an are as it is often expressed), then the various composites of at, a2, all equal to each other. This is the general associative law, which we now prove.. TH E O It E M 2.2. Let (a, b) -} a * b be an associative operation in, an are equal. The common value X. Then all composites of a,, a2, will be written as a,a2 Proof. We use weak induction to define a particular composite a". ofa,,a2,,a,iforn> l: IIis I'a; = a,, II,=,+'a; _ (IIt: ac) a"+1. 74 [lie Natural Number Sequence and its Generalizations I C H A P. 2 Now we prove that for all m,n > 1, (A) (11 a,) * (IT, an+j) = II; +mai. Let n be fixed; we prove by weak induction on in that the relation holds for all in > 1. It is true for in = I by definition. Assume that it is true for in and consider the case in + 1. We have (11;a,) * II", 'a., j = nia, * ((lira" I,) * an,m+,) _ ((n;a,) * +j) * an I In+I n-1 m as required. Thus (A) is valid for all in, n > I. This property of the particular composite defined is used to prove next by strong induction on n that any composite of a,, a,,, n > 1, is equal to 1l a,. Clearly this is true for n = 1. Assume it true for all composites of r elements of X with r G n, and consider any composite associated with (a,, a,, 1,). By definition it is a composite b, a,) and c is a *
c, where b is a composite associated with (a,, a2, then b = composite associated with (a,,,, a,-F2, lira, by the induction hypothesis, c = a,, 1,, and b * c = 1141"'a; by time definition of Ilia,. The proof is then complete for this case. Otherwise, r < it and by the induction hypothesis, a. I,). If r = it, b = Ma;, c = ri; 1' -'a,+; Then b * c = 1I; a; by (A). It follows that all composites of a,, a2,, a are equal, each being equal to 11 a;. Theorem 1.2 can be extended to the following result. THEOREM 2.3. Let B be any rnonerrmpty set and c a given function on 1;"--' into B for it > 2. Let g be any function on N X B" into B. Then there exists exactly one function k: N X B"-' -+ B such that k(p) x2,..., xn) = c(x2,..., xn), k(x', x2,..., Xn) = g(X, k(x, x''2,..., xn), x2,... x,,) The resulting function k is said to be obtained from c and g by primitive recursion. One may think of the earlier theorem as being the special, xn are absent; thus we case which results when all "parameters" X2,. shall also say that the function k of Theorem 1.2 is obtained by primitive 2.2 1 Proof and Definition by Induction 75 recursion. Hints for a direct proof of Theorem 2.3 are given in an exercise. There is considerable interest in the class of number-theoretic functions (that is, functions on N"' into N where p > 1), which can be defined by induction in an elementary way. One motivation for this lies in the possibility of computing values of such functions by purely mechanical means-by a set of instructions which require no "creative" thought in their execution. The operation of primitive recursion enters naturally into such considerations, since if c and g are number-theoretic functions which are computable by mechanical means, then the same appears to be true of the function k obtained from c and g by
primitive recursion. Another operation which appears to produce computable functions from computable functions is that of composition in the following extended sense of our earlier usage of this terra: The function h: N" -j N is obtained by composition from functions f: N" -- N and gc:N"-} N, i = 1, 2,,m,if h(xi,., x") = f(gi(xi,..., x"),..., 9,,,(x,,..., x.,)) g,, The function h obtained in this way will sometimes be written as, g,"). If we specify an initial supply of functions which are judged to be computable, then all functions obtainable by the operations of'composition and primitive recursion should be of the same sort. Such considerations may be taken as motivating the definition of the following class of functions. As the initial supply of functions we take those of the following three types. (I) The successor function S on N : S(x) = x'. (II) The constant functions Cq where CQ(xr,, x,,) = q, n = 1, 2, (III) The identity functions 11,n: N" --} N, where 1 <i <nandn= 1,2, Ui,(x,,..., x,,) = xr,. We next define a primitive recursive derivation to be a finite se, fR of functions, such that any member of the sequence quence fo, f,, is either an initial function or else is obtained from preceding members of the sequence by composition or primitive recursion. Then the class that we have in n-rind, the primitive recursive functions, are those functions f such that there is a primitive recursive derivation whose final 76 Vie Natural Number Sequence and its Generalizations I C t t A r. 2 member is J. This class contains all the numerical functions which one ordinarily encounters, as well as others. Some examples follow. EXAMPLES 2.3. Addition in N is a primitive recursive function. A derivation, wherein we have used function values rather than functions in order to assist with the understanding, is SO,) = y', U2,(-, z, y) = Z, Ax, Z, y) = s(Uz (x, z, y)) - z', U11(y) = y, J a(0, y) = Ui (.y

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