Let $X$, $Y$, and $Z$ be sets.

1. Every \reftext{def:bijection-sets-2026a}{bijection} $u:X\to Y$ is injective; that is, $u(x)=u(x')$ implies $x=x'$ for all $x,x'\in X$.
2. If $u:X\to Y$ and $v:Y\to Z$ are bijections, then the map $w:X\to Z$ defined by $w(x)=v(u(x))$ is a bijection.
3. If $u:X\to Y$ is a bijection and $A\subseteq X$, then, writing $u(A)=\{u(a): a\in A\}$, the restriction of $u$ to $A$ is a bijection from $A$ onto $u(A)$.