CBSE Class 11 Maths Notes: Sets – Algebra of Sets and Power Set

Algebra of Sets:

Idempotent Law: For any set A,

• A ∪ A = A
• A ∩ A = A

Identity Law: For any set A,

• A ∪ ϕ = A
• A ∩ U = A

Commutative Law: For any two sets A and B

• A ∪ B = B ∪ A
• A ∩ B = B ∩ A

Associative Law: For any three sets A, B and C

• ≤ A ∪ B) ∪ C = A ∪ ≤ B ∪ C)
• A ∩ ≤ B ∩ C) = ≤ A ∩ B) ∩ C

Distributive Law: For any three sets A, B and C

• A ∪ ≤ B ∩ C) = ≤ A ∪ B) ∩ ≤ A ∪ C)
• A ∩ ≤ B ∪ C) = ≤ A ∩ B) ∪ ≤ A ∩ C)

De Morgan’s Law: For any two sets A and B

• ≤ A ∪ B)¢ = A¢ ∩ B¢
• ≤ A ∩ B)¢ = A¢ ∪ B¢

POWER SET

The set of all subsets of a given set A is called the power set A and is denoted by P ≤ A). P ≤ A) = {S: S ⊆ A}

For example, if A = {1, 2, 3}, then

P≤ A) = { ϕ,{1},{2},{3},{1},{1,2},{1,3},{2.3},{1,2,3}}

Clearly, if A has n elements, then its power set P≤ A) contains exactly 2ⁿ elements.

Some More Results:

• n ≤ set of elements neither in A nor in B) = n ≤ A¢ ∩ B¢) = n≤ A ∪ B)¢ = n ≤ U) – n ≤ A ∪ B)
• n ≤ A¢ ∪ B¢) = n≤ A ∩ B)¢ = n ≤ U) – n ≤ A ∩ B)
• n ≤ A Δ B) = n [≤ A – B) ∪ ≤ B – A)] = n [≤ A ∩ B¢) ∪ ≤ A¢ ∩ B)] = n ≤ A)         + n ≤ B)         – 2n≤ A ∩ B).

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