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 $ba^{-1} a x = bx$ so $f(x) \in bH$.

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 $b h \in b H$, we want to find $x \in aH$ such that $f(x) = bh$. Let $x = a h$ to obtain $f(x) = b a^{-1} a h = b h$, as required.