TITLE: Cardinality Proof Problem
QUESTION [0 upvotes]: Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A.
I know that there is only an injection from A to B. I'm having trouble showing that set C which is a subset of B has the same cardinality as set A.

REPLY [2 votes]: Since $\text{card}(A)\leq \text{card}(B)$, there is an injection $f\colon A\to B$. Then $f\colon A\to f(A)$ is a bijection, and $f(A)\subseteq B$.
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