Miscellaneous Examples Example Show that the set of letters needed to spell “ CATARACT ” and the set of letters needed to spell “ TRACT” are equal. Solution Let X be the set of letters in “CATARACT”. Then X = { C, A, T, R } Let Y be the set of letters in “ TRACT”. Then Y = { T, R, A, C, T } = { T, R, A, C } Since every element in X is in Y and every element in Y is in X.
It follows that X = Y. Example List all the subsets of the set { – , , }. Solution Let A = { – , , }. The subset of A having no element is the empty set φ .
The subsets of A having one element are { – }, { }, { }. The subsets of A having two elements are {– , }, {– , } ,{ , }. The subset of A having three elements of A is A itself. So, all the subsets of A are φ , {– }, { }, { }, {– , }, {– , }, { , } and {– , , }.
SETS Example Show that A ∪ B = A ∩ B implies A = B Solution Let a ∈ A. Then a ∈ A ∪ B. Since A ∪ B = A ∩ B , a ∈ A ∩ B. So a ∈ B.
Therefore, A ⊂ B. Similarly, if b ∈ B, then b ∈ A ∪ B. Since A ∪ B = A ∩ B, b ∈ A ∩ B. So, b ∈ A.
Therefore, B ⊂ A. Thus, A = B Example For any sets A and B, show that P ( A ∩ B ) = P ( A ) ∩ P ( B ). Solution Let X ∈ P ( A ∩ B ). Then X ⊂ A ∩ B.
So, X ⊂ A and X ⊂ B. Therefore, X ∈ P ( A ) and X ∈ P ( B ) which implies X