What is the difference between subset and proper subset?

A is a subset of B (A⊆B) if every element of A is also in B — this includes the case A = B. A is a proper subset of B (A⊂B) if every element of A is in B AND A ≠ B (A must have fewer elements). Every set is a subset of itself, but not a proper subset of itself.

How many subsets does a set with 4 elements have?

A set with n elements has 2ⁿ subsets. For n = 4: 2⁴ = 16 subsets. This includes the empty set and the set itself. Number of proper subsets = 2ⁿ − 1 = 15.

State De Morgan's Laws and verify with an example.

De Morgan's Laws: (1) (A∪B)' = A'∩B' — the complement of the union equals the intersection of complements. (2) (A∩B)' = A'∪B' — the complement of the intersection equals the union of complements. Example: U={1,2,3,4,5}, A={1,2,3}, B={3,4,5}. A∪B={1,2,3,4,5}, (A∪B)'=∅. A'={4,5}, B'={1,2}, A'∩B'=∅. ✓

Types of Sets and Set Operations

Sets • Class 11 Mathematics • NCERT • CBSE

Types of sets: Empty set (∅), Finite, Infinite, Subset (A⊆B), Power set P(A). Operations: Union A∪B (all elements), Intersection A∩B (common elements), Complement A' (not in A). De Morgan's Laws: (A∪B)' = A'∩B'; (A∩B)' = A'∪B'.

Key Formulas

n(A∪B) = n(A) + n(B) − n(A∩B)
Number of subsets = 2ⁿ
(A∪B)' = A'∩B' (De Morgan's Law 1)
(A∩B)' = A'∪B' (De Morgan's Law 2)

Frequently Asked Questions

What is the difference between subset and proper subset?: A is a subset of B (A⊆B) if every element of A is also in B — this includes the case A = B. A is a proper subset of B (A⊂B) if every element of A is in B AND A ≠ B (A must have fewer elements). Every set is a subset of itself, but not a proper subset of itself.
How many subsets does a set with 4 elements have?: A set with n elements has 2ⁿ subsets. For n = 4: 2⁴ = 16 subsets. This includes the empty set and the set itself. Number of proper subsets = 2ⁿ − 1 = 15.
State De Morgan's Laws and verify with an example.: De Morgan's Laws: (1) (A∪B)' = A'∩B' — the complement of the union equals the intersection of complements. (2) (A∩B)' = A'∪B' — the complement of the intersection equals the union of complements. Example: U={1,2,3,4,5}, A={1,2,3}, B={3,4,5}. A∪B={1,2,3,4,5}, (A∪B)'=∅. A'={4,5}, B'={1,2}, A'∩B'=∅. ✓

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