Injectivity Surjectivity and Inverses

Theorem

A function $S:V\to W$ has an inverse $T:W\to V$ if and only if $S$ is Injective and Surjective

Proof

Proving $⟹$

Suppose $S:V\to W$ has an inverse $T:W\to V$ We want to show that $S$ is injective. Suppose $S(x)=S(y)$ Apply $T$ to get $T(S(x))=T(S(y))⟹x=y$ as $T$ is the inverse of $S$ We want to show that $S$ is surjective Pick $w\in W$, we have $S(T(w))=w$ and so $w$ is a output of $S$ Thus, we have proved $S$ is both injective and surjective

Proving $⟸$

Suppose $S$ is injective and surjective If $S(v)=w$, then we define $T(w)=v$.

• For this $T(w)=v$ to exist, it needs to be well defined.
  • Note $T$ is defined for all $w\in W$ as every $w\in W$ is an output of $S$ ($T(w)=v$ for all $w\in W$) by Surjectivity
  • Note that $T$ is a function because every $w\in W$ is the output of a UNIQUE $v\in V$. If we had $S(v_1)=S(v_2)=w⟹v_1=v_2$ by Injectivity of $S$
• Therefore, $T$ is well defined for all $w\in W$ and $T(w)$ is unique for each $w\in W$ Check that $T(w)$ is the inverse
• $T(S(v))=T(w)=v$
• $S(T(w))=S(v)=w$ Thus, $T(w)$ is the inverse of $S$ by defn of inverse