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The Problem:

Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions.

(a) Show that if $g \circ f$ is injective, then so is $f$.

(b) If $g \circ f$ is surjective, must $g$ be surjective?

Where I Am:

So, I really have trouble with these, for some reason. I can draw pictures and make sense of the problems, but writing down proofs is very difficult for me.

Basically, for (a), I ended up with some complicated statement involving an implication implying another implication and then tried to derive a contradiction. It just got so convoluted that I couldn't make sense of it anymore, and I know there's a quick, elegant way to show it.

For (b), I know that $g$ need not be surjective. Once again, though, proving it directly from definitions has given me a bit of a headache.

Any help here would be appreciated. Thanks in advance.

The Proofs!

Ok, I did it. Thanks for the help, everyone! Let me know if there's anything wrong with these proofs, or if they could bet any better.

(a) Suppose $f$ is not injective. Then $$ f(x_1)=f(x_2) \implies x_1 \ne x_2 \text{ }(*).$$ Let $f(x_1)=y_0=f(x_2)$ and let $g(y_0)=z_0$. Then $$ (g \circ f)(x_1) = (g \circ f)(x_2) = g(y_0) = z_0. $$
Since $g \circ f$ is injective, $$ (g \circ f)(x_1) = (g \circ f_2) \implies x_1 = x_2. $$ However, this contradicts $(*)$. Therefore, $f$ must be injective.

(b) Suppose $g$ is not surjective. Then

$$ \forall y \in Y, \exists z \in Z \text{ such that } g(y) \ne z \text{ }(**).$$

Since, $g \circ f$ is surjective,

$$ \forall z \in Z, \exists x \in X \text{ such that } g(f(x)) = z \text{ } (***). $$

Let $f(x) = y$. Then,

$$ g(f(x)) = g(y) = z. $$

Because of $(***)$, this is true for all $z \in Z$, which contradicts $(**)$. Therefore, $g$ must be surjective.

thisisourconcerndude
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5 Answers5

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(a) Let $f(x_1)=f(x_2)$. Then, $g(f(x_1))=g(f(x_2))$, but since $g\circ f$ is injective...

(b) $g(Y)\supseteq (g\circ f)(X)=g(f(X))$. Hence, if $(g\circ f)(X)=Z$...

  • @:G. Sassatelli ,here is my approach for part b)please correct me if i am wrong $\text{Assume that },, x\epsilon X,,y\epsilon Y,,z\epsilon Z,,. \text{As}, g \circ f,is,,surjective,\Rightarrow g\circ f\left ( x \right )=z. But,, g\circ f \left ( x \right )\subseteq g\left ( y \right )\Rightarrow g\left ( y \right )=z\Rightarrow g,,is,,surjective$ – sourav_anand Mar 26 '17 at 07:21
  • @sourav No, you are mixing the notations for equality of elements and set inclusion. Moreover, you are misusing quantifiers. You either prove that, since for any $z\in Z$ there is $x$ such that $g(f(x))=z$, then $f(x)$ is a valid preimage of $z$ under $g$, or you use my hint about the double inclusion $Y\supseteq g(Y)\supseteq g(f(X))=Y$. –  Mar 26 '17 at 08:36
4

Suppose that $f$ is not injective, then there are $x,y$ such that $y\neq x$ and $f(x)=f(y)$, then we have $g\circ f(x)=g(f(x))=g(f(y))=g\circ f(y)$ which means that $g\circ f$ is also not injective. then by contrapositive we get $g\circ f$ injective $\implies f$ injective as well.

Studzinski
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3

Hint: In each case the contrapositive is obvious.

Bernard
  • 175,478
2

You're probably thinking too hard. For the first one, if $f$ weren't injective, then two elements of $X$ go to the same $y\in Y$. That can only go to one thing in $Z$ so now the composition $g\circ f$ can't be injective after all.

You can reason similarly for surjectivity of $g\circ f$ implying surjectivity of $g$

As an additional exercise, can you see why these statements don't work of you try to prove them for $g$ and $f$, respectively?

If you do it directly, think like this. If $f(x_1) = f(x_2),$ then $g(f(x_1))$ had better equal $g(f(x_2))$, so by injectivity of the composition ...

Neal
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0

(a) Proof:

Since the composite function $g\circ f$ is injective, we have $(g\circ f)(x_{1})=(g\circ f)(x_{2})\implies x_{1}=x_{2}.$ Thus, it follows that $f$ must be injective; otherwise, it would contradict the fact that $(g\circ f)$ is. $\square$

(b) Proof:

If $g \circ f$ is surjective, $\forall z\in Z,\exists x\in X : (g\circ f)(x)=z.$ Let $y=f(x)$, so $y\in Y$ (since $f:X\to Y$). Then, $g(y)=g(f(x))=(g\circ f)(x)=z$, thus, $g$ is surjective. $\square$