I'm working on some exercise in Fraleigh's "A First Course in Abstract Algebra" and one of them involves permutations under which the image of a certain set is a subset (proper or improper) of the set itself.
Let $A$ be a set and $B$ a subset of $A$, and let $\sigma \in S_A$. Is it ever possible for such $A,B,\sigma$ to exists such that $\sigma[B]\subset B$?
Is my below argument sound?
Since $\sigma$ is one-to-one, $\sigma[B]$ should have as many elements as $B$. Hence, if $\sigma[B]$ is a subset of $B$, it must be an improper subset, as a proper subset would require that $|\sigma[B]| < |B|$. That is, $\sigma[B] = B$.