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Question 25126: Let f:X to Y and g:Y to W such that g.f is injective.
a) Show f is injective.
b) Give an example of functions f and g for which g.f is injective but g is not injective.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Let f:X to Y and g:Y to W such that g.f is injective.
a) Show f is injective.
b) Give an example of functions f and g for which g.f is injective
but g is not injective.
a) f(x) = f(y) --> g(f(x)) = g(f(y)) --> g.f(x) = g.f(y)
--> x = y (since g.f is injective)
Hence, f is injecitve.
b) Set f:{1} --> {1} to be f(x) = x (identity map) and
g:{1,2} --> {2} to be g(x) = 2 for x =1,2.
We see that g.f:{1}--> {2 } is injective, but g is not.
Kenny
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