Mapping/Composition/Injective surjective bijection/Fact/Proof/Exercise

Let ${}L,\,M$ and ${}N$ denote sets and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappings with the composition

$G\circ F\colon L\longrightarrow N.$

Show the following properties.

1. If ${}F$ and ${}G$ are injective, then also ${}G\circ F$ is injective.
2. If ${}F$ and ${}G$ are surjective, then also ${}G\circ F$ is surjective.
3. If ${}F$ and ${}G$ are bijective, then also ${}G\circ F$ is bijective.

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