How to show that if $P\subseteq Q$ are finite sets, and $\#P=\#Q$, then $P=Q$?

Question

Let $P, Q$ two finite sets such that : $$P \subset Q$$ $$\#P = \#Q$$

How do you show that $P = Q$ ?

I don't see how I can show that $Q \subset P$

Answer 1

If $Q$ is not contained in $P,$ then there must exist an element $x \in Q$ such that $x \notin P.$ This shows that $\#P < \#Q,$ contradicting the given hypothesis.

Answer 2

HINT: First note that there is an injective function from $Q$ into $Q$ whose range is $P$. Then use the fact that $Q$ is finite.

Answer 3

Induction on $\#Q$.

Base case: $\#P=\#Q=0$. Then $P=\emptyset=Q$.

If $\#P=\#Q=n+1$ then $P\neq\emptyset$. For some $x\in P\subseteq Q$ define $Q'=Q-\left\{ x\right\} $ and $P'=P-\left\{ x\right\} $

Then $\#P'=\#Q'=n$ with $P'\subseteq Q'$ so by induction $P'=Q'$ and consequently $P=P'\cup\left\{ x\right\} =Q'\cup\left\{ x\right\} =Q$.