Let P(A) denote the power set of A. Analyze the given proof and comment on the...

Transcribed Image Text:

Let P(A) denote the power set of A. Analyze the given proof and comment on the correctness of the claim and the proof. Claim: There exist finite sets A, B such that A P(B) = P(A B) where x denotes the Cartesian Product. Proof: 1. Let |A| =n, |B| = k. 2. Let | P(B) = 2^k 3. A x B=nk and hence |P(A B)| = 2nk 4. Then |A P(B)| = n2k. 5. Since |A P(B)| therefore the statement is false. = n2k 2nk = |P(A B)|, Statement is false and proof does not account for smaller sets A, B Statement is true and proof is correct Statement is true but proof is incorrect Statement is false but proof is correct Let P(A) denote the power set of A. Analyze the given proof and comment on the correctness of the claim and the proof. Claim: There exist finite sets A, B such that A P(B) = P(A B) where x denotes the Cartesian Product. Proof: 1. Let |A| =n, |B| = k. 2. Let | P(B) = 2^k 3. A x B=nk and hence |P(A B)| = 2nk 4. Then |A P(B)| = n2k. 5. Since |A P(B)| therefore the statement is false. = n2k 2nk = |P(A B)|, Statement is false and proof does not account for smaller sets A, B Statement is true and proof is correct Statement is true but proof is incorrect Statement is false but proof is correct