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If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$.

Proof:

(a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not always well defined.

(b) Therefore the identity function $f: A \rightarrow B$ defined by $f(x) = x$ is only an injection. Hence by theorem on the textbook, $|A| \le |B|$.

Thank you for your time and gracious helps. ~MA