Problem 5.58. Supposef XY and g : Y Z are functions If g of is one-to-one, prove that fmust be one-to-one 2. Find an example where g o f is one-to-one, but g is not one-to-one

Accepted Solution

Answer with explanation:We are given two functions f(x) and g(y) such that:   f : X → Y  and  g: Y → ZNow we have to show:If gof is one-to-one then f must be one-to-one.Given:gof is one-to-oneTo prove:f is one-to-one.Proof:Let us assume that f(x) is not one-to-one .This means that there exist x and y such that x≠y but f(x)=f(y)On applying both side of the function by the function g we get:g(f(x))=g(f(y))i.e. gof(x)=gof(y)This shows that gof is not one-to-one which is a contradiction to the given statement.Hence, f(x) must be one-to-one.Now, example to show that gof is one-to-one but g is not one-to-one.Let A={1,2,3,4}  B={1,2,3,4,5} C={1,2,3,4,5,6}Let f: A → Bbe defined by f(x)=xand g: B → C be defined by:g(1)=1,g(2)=2,g(3)=3,g(4)=g(5)=4is not a one-to-one function.since 4≠5 but g(4)=g(5)Also, gof : A → Cis a one-to-one function.