to prove the statement: If A B and card(A) = card(B), then A=B. This proof is valid for finite sets.

Welcome back — missed you this week. The question asks to prove the statement: If A B and card(A) = card(B), then A=B. This proof is valid for finite sets. Step 1: State the given conditions. We are given two finite sets A and B such that: • A B (This means every element of A is also an element of B). • card(A) = card(B) (This means A and B have the same number of elements). Step 2: Define set equality. To prove that A=B, we must show that A B and B A. We are already given A B, so we only need to prove B A. Step 3: Prove B A by contradiction. Assume, for the sake of contradiction, that B A. If B A, it means there exists at least one element x such that x B and x A. Since A B, all elements of A are already included in B. If there is an element x in B that is not in A, then B contains all the elements of A plus at least one additional element (x). This implies that the number of elements in B must be strictly greater than the number of elements in A. Thus, we would have card(B) > card(A). Step 4: Identify the contradiction. The conclusion from Step 3, card(B) > card(A), directly contradicts our given condition that card(A) = card(B). Therefore, our initial assumption that B A must be false. This means that B A must be true. Step 5: Conclude. Since we have established both A B (given) and B A (proven), we can conclude that: A=B This completes the proof of the auxiliary hypothesis. Send me the next one 📸