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# Am I on the right track to proving this statement about composite function?

2018-03-18 22:43:16

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$.

Now I think this statement is true:

$\forall f \in F , \exists g\in F$ so that $(g \circ f)(1) =2$

I suppose $f \in F$, and I need to show that $\exists g\in F$ so that $(g \circ f)(1) = 2$

Do I just make an example of $g$ so that $(g \circ f)(1) = 2$ ?

Like $f(1) = y$ and then $g (y)=2 ? You could simply design a function$g$such that$g(x)=2$for all$x\in S$. Now since$f\in F$, we know$f(1)\in S$, hence the composition is 2. As f is arbitrary, we have the desired result. Yes, you just define a function$g$which takes$f(1)$to$2$and then define the rest if images to make$g$a well defined function from$S$to$S$• You could simply design a function$g$such that$g(x)=2$for all$x\in S$. Now since$f\in F$, we know$f(1)\in S$, hence the composition is 2. As f is arbitrary, we have the desired result. 2018-03-18 23:40:01 • Yes, you just define a function$g$which takes$f(1)$to$2$and then define the rest if images to make$g$a well defined function from$S$to$S\$

2018-03-19 00:30:22