| --- |
| 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). |
