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Homework Statement
If f:A->B is a bijection and g:B->C is a bijection, show that g[itex]\circ[/itex]f is a bijection from A->C.
Homework Equations
A function is a bijection from A to B when it is both a surjection and an injection from A to B.
The Attempt at a Solution
Suppose f is a bijection from A to B and g is a bijection from B to C. We want to show that h=g[itex]\circ[/itex]f is a bijection from A to C. Since f is a bijection from A to B we have that the Rng(f)=B. Now since g is a function from B to C we have that Rng(f)=Dom(g). Let f(x)=y. Then y[itex]\in[/itex]Rng(f). Thus y[itex]\in[/itex]Dom(g). Now since we have that g is a bijection from B to C, g(y) is a bijection from B to C because y[itex]\in[/itex]B but y=f(x) so g(f(x)) is a bijection. Now f(x) was a bijection from A to B so g(f(x)) is a bijection from A to B to C. In particular, g(f(x)) is a bijection from A to C. By definition we have that g(f(x))=g[itex]\circ[/itex]f. Thus h=g[itex]\circ[/itex]f is a bijection from A to C.
I would appreciate it if someone could read over my proof and give me some feedback on it. Thanks a lot!