5. Let X, Y and Z be sets. Let f : X ! Y and g...

5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z functions.

(a) (3 Pts.) Show that if g f is an injective function, then f is an injective function.

(b) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is

injective but g is not injective.

(d) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is

surjective but f is not injective.

Solutions

Expert Solution

For part "d" of the question, I know it has asked for mappings f and g such that got is surjective but f is injective, and I have provided that, but the same example is also such that f is also not surjective (to maintain symmetry with part "b").

So, if g°f is an injective function, then so is f.

This is an example where g°f is injective, and g is not injective.

So if g°f is a surjective function, then so is g.

This is an example where g°f is surjective, and f is neither injective nor surjective.

Colby Messinger answered 1 year ago

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