Let A and B be two arbitrary sets and let ^ denote the set intersection operator

Question

Let A and B be two arbitrary sets and let ^ denote the set intersection operator and - denote the set difference operator. Show, by a two-sided proof, that (A-(A-B)) = (A^B)

Explanation / Answer

let x be in A-(A-B) ==> that x is in A and x is not in A-B this is possible when x is in both A and B. ==> x is in A^B. so A-(A-B) is subset of A^B. .................... 1 now let x be in A^B ==> x is in both A and B. so x is not in A-B ==> x is present in A-(A-B) [because x is in A and not in A-B] hence A^B is sub set of A-(A-B) ................. 2 from 1 and 2 , we get A-(A-B) = A^B

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