MHB Compose Functions: True/False?

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The discussion centers on the properties of function composition, specifically regarding injective, surjective, and bijective functions. It is established that if both f and g are injective, surjective, or bijective, then their composition g∘f retains those properties. However, it is clarified that if g∘f is bijective, it does not necessarily imply that either f or g must also be bijective. The main point of contention is whether f∘g equals g∘f when A, B, and C are equal sets; the consensus is that equality of sets does not guarantee the equality of function compositions. The conclusion emphasizes that function equality requires matching outputs for all inputs, not just matching domains and codomains.
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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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