I was reading the answer of letter (a) of this question:
The sum of two sets and the disjoint union.
but I did not understand it.
My questions are:
1-what is the Definition of injective maps in Category theory? and what is the relation of this definition to our ordinary definition (if $f(x) = f(y)$ then $x=y$)?
2- Also in the solution, I did not understand this statement "As the only map $\emptyset \mapsto X$ is injective, WLOG $S\neq \emptyset$.
We use the following (easy) claim : if $v \circ f$ is injective, then $f$ is injective. ", could anyone explain it for me please? from where this claim "We use the following (easy) claim : if $v \circ f$ is injective, then $f$ is injective." came?
In the question, all the objects are sets and the maps are ordinary functions. The usual definition of injective applies here. If we were in a more general setting, we could ask that the morphism be a monomorphism. You should prove to yourself that an ordinary function is monomorphic if and only if it's injective.
We're trying to prove that a map $S \to X$ is injective. Since every set is empty or non-empty, we only need to check those two cases. For the first case, $S = \emptyset$. Since every map $\emptyset \to X$ is injective, we're done. Then we proceed with the meat of the argument with the $S \neq \emptyset$ case.
Have you tried proving this? It's a fairly straightforward application of the the definition of injective.