- #1

lemonthree

- 51

- 0

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.