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.