I struggling a little bit with sufficient and necessary conditions. I have to find sufficient and necessary conditions for the composition of two functions $f \circ g$ being injective, surjective or both.
As Definition for $f \circ g$ I've got $g(f(x))$. We say $A$ is sufficient for $B$, if $A$ implies $B$ $(A \Rightarrow B)$ and we say $A$ is necessary for B, if $B$ can't hold without $A$, $B \Rightarrow A$. $A$ is necessary and sufficient if $A \Leftrightarrow B$ holds.
By drawing some picture with different cases I found the following
- $f$ injective + $g$ injective $\Rightarrow f \circ g$ injective
- $f$ surjective + $g$ surjective $\Rightarrow f \circ g$ surjective
- $f$ bijective + $g$ bijective $\Rightarrow f \circ g$ bijective
And the other way around:
- $f \circ g$ injective $\Rightarrow$ $f$ injective + $g$ can be both
- $f \circ g$ surjective $\Rightarrow$ $f$ can be both + $g$ surjective
- $f \circ g$ bijective $\Rightarrow$ $f$ injective + $g$ surjective
My solution for sufficient and necessary conditions:
- $f \circ g$ injective: f injective + g injective are sufficient, f injective is necessary
- $f \circ g$ surjective: f surjective + g surjective are sufficient, g surjective is necessary
- $f \circ g$ bijective: f bijective + g bijective are sufficient, f injective and g surjective are necessary
Are my ideas correct so far and are there condicions which are necessary and sufficient at the same tie? I would say "no".
Any hints and ideas are very welcome.