If $f:W\rightarrow X$, $g:X\rightarrow Y$, and $h:Y\rightarrow Z$, does $h \circ (g \circ f) = (h \circ g) \circ f$? How can I justify this?
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2Your question is unclear. What do you mean with "does h(g(x)) of f = h(g(f(x))"? – Newb Dec 31 '13 at 23:52
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is h o (g o f) = (h o g) o f Does that explain it? – lifeofjuds Dec 31 '13 at 23:53
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1Do you mean to ask why is $h\circ(g\circ f)=(h\circ g)\circ f$? – hmakholm left over Monica Dec 31 '13 at 23:54
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The answer to the question in the comments is "yes". – vadim123 Dec 31 '13 at 23:54
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Yes - to henning makholm. I just could not figure out how to format that as a question. Vadim123, can you explain why? – lifeofjuds Dec 31 '13 at 23:55
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For every $w\in W$ we have $$ [h\circ (g\circ f)](w) = h([g\circ f](w)) = h(g(f(w)) $$ and $$ [(h\circ g)\circ f](w) = [h\circ g](f(w)) = h(g(f(w)) $$
Since $h\circ (g\circ f)$ and $(h\circ g)\circ f$ have the same value at every possible argument, they are the same function.
hmakholm left over Monica
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and since they have the same domain and codomain (trivial remark, but the question is trivial anyway) – Alexander Grothendieck Jan 01 '14 at 00:17