Welcome back — missed you this week. The question asks to prove the statement: If $A \subset B$ and $\text{card}(A) = \text{card}(B)$, 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: • $A \subset B$ (This means every element of $A$ is also an element of $B$). • $\text{card}(A) = \text{card}(B)$ (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 $A \subset B$ and $B \subset A$. We are already given $A \subset B$, so we only need to prove $B \subset A$. Step 3: Prove $B \subset A$ by contradiction. Assume, for the sake of contradiction, that $B \not\subset A$. If $B \not\subset A$, it means there exists at least one element $x$ such that $x \in B$ and $x \notin A$. Since $A \subset B$, 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 $\text{card}(B) > \text{card}(A)$. Step 4: Identify the contradiction. The conclusion from Step 3, $\text{card}(B) > \text{card}(A)$, directly contradicts our given condition that $\text{card}(A) = \text{card}(B)$. Therefore, our initial assumption that $B \not\subset A$ must be false. This means that $B \subset A$ must be true. Step 5: Conclude. Since we have established both $A \subset B$ (given) and $B \subset A$ (proven), we can conclude that: $$A=B$$ This completes the proof of the auxiliary hypothesis. Send me the next one 📸