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Distinguish between equal sets which have exactly the same members and equivalent sets that may have different members but must have the same cardinality or size. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Every member of a subset of a set is also a member of the set containing it.AâBAâB | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
A proper subset of a set does not contain all the members of the set containing it. There is a least one member of setBBthat is not a member of setAA.AâBAâB | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The number subsets of a finite setAAwithn(A)n(A)members is equal to 2 raised to then(A)n(A)power. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The empty set is a subset of every set and must be included when listing all the subsets of a set. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Understand how to create and distinguish between equivalent subsets of finite and infinite sets that are not equal to the original set. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
A Venn diagram is a graphical representation of the relationship between sets. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
In a Venn diagram, the universal set,UUis the largest set under consideration and is drawn as a rectangle. All subsets of the universal set are drawn as circles within this rectangle. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The complement of setAAincludes all the members of the universal set that are not in setAA. A set and its complement are disjoint sets, they do not share any elements in common. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
To find the complement of setAAremove all the elements of setAAfrom the universal setUU, the set that includes only the remaining elements is the complement of setAA,Aâ²Aâ². | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Determine the complement of a set using Venn diagrams, the roster method and set builder notation. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The intersection of two sets,Aâ©BAâ©Bis the set of all elements that they have in common. Any member ofAAintersectionBBmust be is both setAAand setBB. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The union of two sets,AâªBAâªB, is the collection of all members that are in either in setAA, setBBor both setsAAandBBcombined. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Two sets that share at least one element in common, so that they are not disjoint are represented in a Venn Diagram using two circles that overlap.The region of the overlap is the setAAintersectionBB,Aâ©B.Aâ©B.The regions that include everything in the circle representing setAAor the circle representing setBBor their... | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The region of the overlap is the setAAintersectionBB,Aâ©B.Aâ©B. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
The regions that include everything in the circle representing setAAor the circle representing setBBor their overlap is the setAAunionBB,AâªB.AâªB. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Apply knowledge of set union and intersection to determine cardinality and membership using Venn Diagrams, the roster method and set builder notation. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
A Venn diagram with two overlapping sets breaks the universal set up into four distinct regions. When a third overlapping set is added the Venn diagram is broken up into eight distinct regions. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Analyze, interpret, and create Venn diagrams involving three overlapping sets.Including the blood factors: A, B and RhTo find unions and intersections.To find cardinality of both unions and intersections. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Including the blood factors: A, B and Rh | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
To find unions and intersections. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
To find cardinality of both unions and intersections. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
When performing set operations with three or more sets, the order of operations is inner most parentheses first, then fine the complement of any sets, then perform any union or intersection operations that remain. | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
To prove set equality using Venn diagrams the strategy is to draw a Venn diagram to represent each side of the equality or equation, then look at the resulting diagrams to see if the regions under consideration are identical. If they regions are identical the equation represents a true statement, otherwise it is not tr... | https://openstax.org/books/contemporary-mathematics/pages/1-key-concepts |
Logical statements have the form of a complete sentence and make claims that can be identified as true or false. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Logical statements are represented symbolically using a lowercase letter. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The negation of a logical statement has the opposite truth value of the original statement. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Be able toDetermine whether a sentence represents a logical statement.Write and translate logical statements between words and symbols.Negate logical statements, including logical statements containing quantifiers ofall, some, and none. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Determine whether a sentence represents a logical statement. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Write and translate logical statements between words and symbols. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Negate logical statements, including logical statements containing quantifiers ofall, some, and none. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Logical connectives are used to form compound logical statements by using words such asand, or, and if â¦, then. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A conjunction is a compound logical statement formed by combining two statements with the words âandâ or âbut.â If the two independent clauses are represented byppandqq, respectively, then the conjunction is written symbolically aspâ§qpâ§q. For the conjunction to be true, bothppandqqmust be true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A disjunction joins two logical statements with theorconnective. In, logicoris inclusive. For anorstatement to be true at least one statement must be true, but both may also be true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A conditional statement has the form ifpp, thenqq, whereppandqqare logical statements. The only time the conditional statement is false is whenppis true, andqqis false. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The biconditional statement is formed using the connectiveif and only ifif and only iffor the biconditional statement to be true, the true values ofppandqq, must match. Ifppis true thenqqmust be true, ifppis false, thenqqmust be false. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Translate compound statements between words and symbolic form.ConnectiveSymbolNameandbutâ§â§conjunctionorâ¨â¨disjunction, inclusive ornot~negationifâ¦â¦, then
impliesââconditional, implicationif and only ifââbiconditional | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The dominance of connectives should be applied in the following orderParenthesesNegationsDisjunctions/Conjunctions, left to rightConditionalsBiconditionalsFigure2.18 | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Parentheses | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Negations | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Disjunctions/Conjunctions, left to right | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Conditionals | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Biconditionals | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Determine the true values of logical statements involving negations, conjunctions, and disjunctions.The negation of a logical statement has the opposite true value of the original statement.A conjunction is true when bothppandqqare true, otherwise it is false.A disjunction is false when bothppandqqare false, otherwise ... | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The negation of a logical statement has the opposite true value of the original statement. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A conjunction is true when bothppandqqare true, otherwise it is false. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A disjunction is false when bothppandqqare false, otherwise it is true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.NegationConjunction (AND)Disjunction (OR)pp~p~pppqqpâ§qpâ§qppqqpâ¨qpâ¨qTFT... | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The conditional statement, ifppthenqq, is like a contract. The only time it is false is when the contract has been broken. That is, whenppis true, andqqis false.ConditionalppqqpâqpâqTTTTFFFTTFFT | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The biconditional statement,ppif and only ifqq, it true wheneverppandqqhave matching true values, otherwise it is false.BiconditionalppqqpâqpâqTTTTFFFTFFFT | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to construct truth tables involving conditional and biconditional statements. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Use truth tables to analyze conditional and biconditional statements and determine their validity. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Two statementsppandqqare logically equivalent if the biconditional statement,pâqpâqis a valid argument. That is, the last column of the truth table consists of only true values. In other words,pâqpâqis a tautology. Symbolically,ppis logically equivalent toqqis written as:pâ¡q.pâ¡q. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A logical statement is a tautology if it is always true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
To be valid a local argument must be a tautology. It must always be true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know the variations of the conditional statement, be able to determine their truth values and compose statements with them. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The converse of a conditional statement, ifppthenqq, is the statement formed by interchanging the hypothesis and conclusion. It is the statement ifqqthenpp. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The contrapositive negates and interchanges the hypothesis and the conclusion.ConditionalContrapositiveConverseInverseppqq~p~p~q~qpâqpâq~qâ~p~qâ~pqâpqâp~pâ~q~pâ~qTTFFTTTTTFFTFFTTFTTFTTFFFFTTTTTT | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The conditional statement is logically equivalent to the contrapositive. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The converse is logically equivalent to the inverse. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to construct and use truth tables to determine whether statements are logically equivalent. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
De Morganâs Law for the negation of a disjunction states that,~(pâ¨q)~(pâ¨q)is logically equivalent to~pâ§~q.~pâ§~q. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
De Morganâs Law The negation of a conjunction states that,~(pâ§q)â¡~pâ¨~q.~(pâ§q)â¡~pâ¨~q. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Use De Morganâs Laws to negate conjunctions and disjunctions. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The negation of a conditional statement, ifppthenqqis logically equivalent to the statementppand notqq. Use this property to write the negation of conditional statements. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Use truth tables to evaluate De Morganâs Laws. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The law of detachment is a valid form of a conditional argument that asserts that if both the conditional,pâqpâqis true and the hypothesis,ppis true, then the conclusionqqmust also be true.Law of DetachmentPremise:pâqpâqPremise:ppConclusion:â´qâ´q | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to apply the law of detachment to determine the conclusion of a pair of statements. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Thelaw of denying the consequentis a valid form of a conditional argument that asserts that if both the conditional,pâqpâqis true and the negation of the conclusion,~q~qis true, then the negation of the hypothesis~p~pmust also be true.Law of Denying the ConsequentPremise:pâqpâqPremise:~q~qConclusion:â´~pâ´~p | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form,pâqpâqandqârqâr, then it follows thatpâr.pâr.Chain Rule for Conditional ArgumentsPremise:pâqpâqPremise:qârqârConclusion:â´pârâ´pâr | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
Know how to apply the chain rule to determine valid conclusions for pairs of true statements. | https://openstax.org/books/contemporary-mathematics/pages/2-key-concepts |
ac±bc=a±bcac±bc=a±bc | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
abÃcd=aÃcbÃdabÃcd=aÃcbÃd | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
abÃcd=aÃcbÃdab÷cd=adÃdc=aÃdbÃcabÃcd=aÃcbÃdab÷cd=adÃdc=aÃdbÃc | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃb=aÃbaÃb=aÃb | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃx±bÃx=(a±b)ÃxaÃx±bÃx=(a±b)Ãx | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a÷b=ab=aba÷b=ab=ab | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a2âb2=(aâb)(a+b)a2âb2=(aâb)(a+b) | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃ(b+c)=aÃb+aÃcaÃ(b+c)=aÃb+aÃc | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a+b=b+aa+b=b+a | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃb=bÃaaÃb=bÃa | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃ(bÃc)=(aÃb)ÃcaÃ(bÃc)=(aÃb)Ãc | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a+0=aa+0=a | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃ1=aaÃ1=a | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a+(âa)=0a+(âa)=0 | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aÃ(1a)=1aÃ(1a)=1 | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
anam=an+manam=an+m | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
anam=a(nâm)anam=a(nâm) | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
a0=1a0=1, provided thataâ0aâ0 | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
(aÃb)n=anÃbn(aÃb)n=anÃbn | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
(ab)n=anbn(ab)n=anbn | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
(an)m=a(nÃm)(an)m=a(nÃm) | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
aân=1anaân=1an, provided thataâ0aâ0 | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
ai=a1+dÃ(iâ1)ai=a1+dÃ(iâ1) | https://openstax.org/books/contemporary-mathematics/pages/3-formula-review |
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