# Thread: Proof: cardinalities of sets

1. ## Proof: cardinalities of sets

Let a be the common cardinality of sets A and C. Let b be the size of B. If C is a subset of B, then a<= b.

I understand why this is true, but how do I give a solid proof?

Thanks.

2. Originally Posted by jzellt
Let a be the common cardinality of sets A and C. Let b be the size of B. If C is a subset of B, then a<= b.

I understand why this is true, but how do I give a solid proof?

Thanks.
note that if $C \subseteq B$, then $|C| \le |B|$ (you can prove this by contradiction).

here, since $|A| = |C|$, we have $|A| \le |B|$