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

Answer with explanation: We are given two functions f (x) and g (y) such that:

f : X → Y and g: Y → Z

Now we have to show:

If gof is one-to-one then f must be one-to-one.

Given:

gof is one-to-one

To 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 → B

be defined by f (x) = x

and g: B → C be defined by:

g (1) = 1, g (2) = 2, g (3) = 3, g (4) = g (5) = 4

is not a one-to-one function.

since 4≠5 but g (4) = g (5)

Also, gof : A → C

is a one-to-one function.