SOLUTION: 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.

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
RELATED QUESTIONS
Please help me solve this: 1. In how many ways can 5 people be arranged on a... (answered by stanbon)

This is coming from the University of Houston online textbook, section 3.6 #39, if that... (answered by stanbon)

give an example of three f,g, and h (none of which is a constant function) such that... (answered by richard1234)

If f and g are linear functions such that g(f(x)) = 2x+ 6 and the graph of y = f(g(x))... (answered by stanbon)

Are there any functions defined on real numbers or rational numbers other than zero... (answered by ikleyn)

Provide an example of functions f(x) and g(x) with the limit as x approaches infinity of... (answered by richard1234)

If N represents the set of natural numbers, and the function f: N → N such that... (answered by MathLover1)

a function is given: f(x) = 3x + 12 a. determine the inverse of this function and name... (answered by stanbon)

Let f : A ��> B and g : B ��> C , two bijective functions. Prove that g�f is... (answered by robertb)