Let $H$ be a subgroup of $G$. Then for any two left cosets of $H$ in $G$, there is a bijective function between the two cosets.

Proof

Let $aH,bH$ be two cosets. Define the function $f:aH\to bH$ by $x\mapsto b{a}^{-1}x$.

This has the correct codomain: if $x\in aH$ (so $x=ah$, say), then $b{a}^{-1}ax=bx$ so $f\left(x\right)\in bH$.

The function is injective: if $b{a}^{-1}x=b{a}^{-1}y$ then (pre-multiplying both sides by $a{b}^{-1}$) we obtain $x=y$.

The function is surjective: given $bh\in bH$, we want to find $x\in aH$ such that $f\left(x\right)=bh$. Let $x=ah$ to obtain $f\left(x\right)=b{a}^{-1}ah=bh$, as required.