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k3k3

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## Homework Statement

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|.

## Homework 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|

## 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?