question_answer 62)

Give examples of two functions f : N  Z and g : Z  Z such that gof is injective but g is not injective.  

Answer:

Let f(x) = x and g(x) = |x|       g(?x) = |?x| = |x|              is not one-one i.e. injective.            Also, gof :       Let x1 : x2 be two elements of N such that (gof) (x1)         = (gof ) (x2)       g(f(x1)) = g(f(x2))       g(x1) = g(x2)       |x1| = |x2|       x1 = x2             is injective.