# Proving Surjective and Injective Functions: f and g are Injections

• Lee33
In summary, if f: A→B is a surjection and g:B→C is such that g∘f is an injection, then f and g are both injections. This can be proven by showing that f is injective and g is injective using the given conditions.
Lee33

## Homework Statement

Let f: A→B be a surjection and g:B→C be such that g∘f is an injection. Prove that both f and g are injections

## The Attempt at a Solution

Suppose a = a' then g(f(a)) = g(f(a')) since g∘f is 1-1 we have that f(a) = f(a') hence f is 1-1.

Is this correct?

You assumed that a=a'and proved that f(a) = f(a'), which is simply showing that f is a function, not anything to do with whether f is 1-1. Furthermore, you went from g(f(a)) = g(f(a')) to f(a) = f(a') which seems to assume that g is injective already.

You never used the fact that f is a surjection, which should also tip you off that you missed something.

1 person
Let me redo.

Suppose a,a' ∈ A and that f(a) = f(a') then g(f(a)) = g(f(a')) since g∘f is an injection then we have a = a' therefore f is injective.

Is this correct?

That looks better. Any ideas on what to do for g?

Thanks! For g:

Let b, b' ∈ B. Then since f is onto there exists an a, a' ∈ A such that f(a) = b and f(a') = b'. Suppose g(b)=g(b') then we get g(b) = g(f(a)) = g(f(a'))= g(b') since g∘f is 1-1 then a = a' hence f(a) = f(a') therefore b = b' which means g is 1-1.

Is this correct?

That looks good to me.

1 person
Thank you very much!

## What is a 1-1 function?

A 1-1 function, also known as an injective function, is a function where each element in the domain has a unique element in the range. This means that no two elements in the domain can map to the same element in the range.

## What is an onto function?

An onto function, also known as a surjective function, is a function where every element in the range has at least one corresponding element in the domain. This means that every element in the range is mapped to by at least one element in the domain.

## How can I determine if a function is 1-1?

To determine if a function is 1-1, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not 1-1. Another way is to check if each element in the range has a unique preimage in the domain.

## How can I determine if a function is onto?

To determine if a function is onto, you can check if every element in the range is mapped to by at least one element in the domain. Another way is to check if the range of the function is equal to the codomain.

## Can a function be both 1-1 and onto?

Yes, a function can be both 1-1 and onto. This type of function is called a bijection. It means that each element in the domain has a unique element in the range and every element in the range is mapped to by exactly one element in the domain.

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