SOLUTION: Let f : A ��> B and g : B ��> C , two bijective functions. Prove that g�f is bijective

Question 1103170: Let f : A ��> B and g : B ��> C , two bijective functions. Prove that g�f is bijective
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Let a, b be elements of A such that
g(f(a)) = g(f(b)).
g 1-to-1 ==> f(a) = f(b).
f 1-to-1 ==> a = b.
==> g(f(x)) is 1-to-1 also.-------------(1)
Now let z be an element of C.
g DISABLED_event_onto= => there is y an element of B such that g(y) = z.
f DISABLED_event_onto= => there is x an element of A such that f(x) = y.
===> g(f(x)) = z.
===> g o f is also onto.--------------(2)
Therefore g o f is also bijective, from (1) and (2).
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