Is (h\circ g)\circ f = h\circ (g\circ f)?

[DaDramaQueen]

1. Prove that
$$(h\circ g)\circ f = h\circ (g\circ f)$$

Homework Equations

$$f:A\longmapsto B$$,


$$g:B\longmapsto C$$,


$$h:C\longmapsto D$$

The Attempt at a Solution

$$(h\circ g)\circ f =\{(b,d):d=h(c)\}\circ f$$


$$=\{(b,d):d=h(g(b))\}\circ f$$

I reach there and get stuck to continue

 

[HallsofIvy]

That looks like a cumbersome way approach.

Let "x" be a member of the domain of f such that f(x) is in the domain of g and g(f(x)) is in the domain of h. Then x is in the domain of $h\circ (g\circ f)$ and $h\circ (g\circ f)(x)= h(g(f(x)))$.

Since g(f(x)) is in the domain of h, f(x) is in the domain of $h\circ g$ and $(h\circ g)(f(x))= h(g(f(x))$. QED.