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I am a undergraduate majoring in CS. In preparation for a discrete mathematics exam coming up next week, I am looking through problems I got wrong on the homework. A concept I don't understand are surjections, injections, and bijection. From lecture, for a function to be a bijection, it has to be both an injection and a surjection. So say I proved a function is not a surjection, why couldn't I say that it has to be injection since we know it can't be a bijection by definition?
So my homework problem is in the link below.
Problem 4.26. Let $A$, $B$, and $C$ be sets and let $f\colon B\to C$ and $g\colon A\to B$ be functions. Let $h\colon A\to C$ be the composition, $f\circ g$, that is, $h(x)::=f(g(x))$ for $x\in A$. Prove or disprove the following claims.
- (a) If $h$ is surjective, then $f$ must be surjective.
- (b) If $h$ is surjective, then $g$ must be surjective.
- (c) If $h$ is injective, then $f$ must be injective.
- (d) If $h$ is injective and $f$ is total, then $g$ must be surjective
I got a) True b) False c) True d) False
When the answer is supposed to be a) True b) false c) false d) true
I think the reason why I got them wrong is because I assumed that if a function is not surjective, then it has to be injective and vice versa. Could someone help me understand this concept? That would be much appreciated!