Injective functions

**hebby** · November 19th 2009, 03:55 PM

Let f : A → B et g : B → C be 2 functions

Show that if g ◦ f is injective, then f has to be injective

**Focus** · November 19th 2009, 04:51 PM

This isn't true, take $\cdot ^2:\mathbb{R}\rightarrow[0,\infty)$ and $\sqrt{}:[0,\infty)\rightarrow [0,\infty)$ .

**Jose27** · November 19th 2009, 05:05 PM

Originally Posted by Focus

This isn't true, take $\cdot ^2:\mathbb{R}\rightarrow[0,\infty)$ and $\sqrt{}:[0,\infty)\rightarrow [0,\infty)$ .

Notice that $\sqrt{} \circ \cdot^2$ is not injective.

As for the problem assume $f(x)=f(y)$ then $g(f(x))=g(f(y))$ which implies $x=y$ by the injectivity of $g\circ f$

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