MHB Compose Functions: True/False?

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Which of the following are true? Select all options. Assume that f:A→B and g:B→C.

If f and g are injective, then so is g∘f
If f and g are surjective, then so is g∘f
If f and g are bijective, then so is g∘f
If g∘f is bijective, then so are both f and g
If g∘f is bijective, then so is either f or g
If A=B=C, then f∘g=g∘f.

The first 3 options are true while the next 2 options are false. The only question I have here is regarding the last option "If A=B=C, then f∘g=g∘f."

Why can't it be true? If A = B = C, and they are all equal sets, we know that f:A→B and g:B→C.
So in essence, isn't f∘g=g∘f?
f∘g= f(g(x)) = f(C) = f(A) = B
While g∘f = g(f(x)) = g(B) = C
But A = B = C so they are all equal.
Range = codomain = domain anyway.
 
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lemonthree said:
The only question I have here is regarding the last option "If A=B=C, then f∘g=g∘f."
Then it makes sense to post only that question.
lemonthree said:
Why can't it be true?
The question is not whether $f\circ g=g\circ f$ can be true, it's whether it must be true under the given assumptions.
lemonthree said:
If A = B = C, and they are all equal sets, we know that f:A→B and g:B→C.
More than that, we are given that $f:A\to B$ and $g:B\to C$ even if $A=B=C$ does not hold.

(A relevant joke. A tourist asks a local: "If I go down this street, will there be a railway station? The local replies, "The station will be there even if you don't go down that street".)

lemonthree said:
f∘g= f(g(x)) = f(C)
$f(g(x))$ cannot equal $f(C)$ because the former is an element of the codomain of $f$ while $f(C)$, if defined, is a subset of the codomain of $f$. In any case, equality of two functions $h_1$ and $h_2$ means that $h_1(x)=h_2(x)$ for every $x$ in the domain of $h_1$ and $h_2$ and not just that the domain and codomain of $h_1$ and $h_2$ are the same.
 
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