1. The problem statement, all variables and given/known data If A and B are sets we say that |A|≤|B| if and only if there exists a one-to-one function f:A→B. Prove that if A and B are sets such that A[itex]\subseteq[/itex]B , then |A|≤|B|. 2. Relevant equations Our text does not define this, so the definition comes from my class notes. Definition: Suppose A and B are sets. Then |A|≤|B| iff there exists a 1-1 function from A to B. Note: Such an an f, if it exists, may or may not be onto. If f is also onto, then |A|=|B| 3. The attempt at a solution Since A[itex]\subseteq[/itex]B, then either A is B or all of A is in B. Hence, there exists an injection f such that f:A→B defined by f(a)=a for all a in A. Since f is not a onto B, then |A| < |B| If f is a bijection, then, by definition, it follows that |A|=|B|. Does this proof make sense?