source: ncert
topic: sets
class_level: class_11
chapter: ch1
difficulty: intermediate
Class 11 | Ch1: Sets SET: well-defined collection of distinct objects. Representation: Roster (list) {1,2,3} or Set-builder {x: condition}. TYPES OF SETS: Empty/Null set: no elements. phi or {}. Singleton: exactly one element. Finite set: countable elements. Infinite set: uncountable. Universal set U: contains all elements under consideration. Subset: A is subset of B if every element of A is in B. A⊆B. Proper subset: A⊂B means A⊆B and A≠B. Power set P(A): set of ALL subsets of A. If n(A)=k, then n(P(A))=2^k. SET OPERATIONS: Union A∪B: elements in A OR B (or both). Intersection A∩B: elements in BOTH A and B. Difference A-B: elements in A but NOT in B. Complement A': elements in U but NOT in A. A'=U-A. Symmetric difference A△B = (A-B)∪(B-A). LAWS (De Morgan's): (A∪B)'=A'∩B'. (A∩B)'=A'∪B'. VENN DIAGRAMS: circles inside rectangle (universal set). CARDINAL NUMBER FORMULA: n(A∪B)=n(A)+n(B)-n(A∩B). n(A∪B∪C)=n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C). SOLVED EXAMPLES: Example 1: A={1,2,3,4,5}, B={2,4,6,8}. A∪B={1,2,3,4,5,6,8}. A∩B={2,4}. A-B={1,3,5}. B-A={6,8}. n(A∪B)=5+4-2=7. Verified: 7 elements in A∪B. Example 2: n(A)=20, n(B)=25, n(A∪B)=40. Find n(A∩B). n(A∩B)=20+25-40=5. Example 3: P({a,b})={{},{a},{b},{a,b}}. n=4=2^2. COMMON MISTAKES: {0} and {} are DIFFERENT. {0} has one element (zero). {} is empty. Empty set {} is subset of EVERY set. A⊆A always (every set is subset of itself).