Science:Math Exam Resources/Courses/MATH220/December 2010/Question 04 (a)

MATH220 December 2010

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Question 04 (a)
Let $f:A\to {B}$ and $g:B\to {A}$ be functions so that $g\circ {f}$ is an injection. Prove that $\displaystyle {}f$ must be injective.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that a function $\displaystyle h:A\to B$ is injective if and only if for all values $\displaystyle a,a'\in A$, we have that $\displaystyle h(a)=h(a')$ implies that $\displaystyle a=a'$. Use this definition for $\displaystyle g\circ f$ and then show that $\displaystyle f$ satisfies this definition.

Hint 2
Try a direct proof, that is, let $a,a'\in A$ be such that $f(a)=f(a')$. How can you use the information about $g\circ f$ to show that $a=a'$?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The function $\displaystyle g\circ f:A\to A$ is given to be injective, meaning that for $\displaystyle a,a'\in A$, we have that if $\displaystyle g\circ f(a)=g\circ f(a')$ then necessarily $\displaystyle a=a'$. To show that $f$ is an injective function, we need to show that for any $\displaystyle a,a'\in A$ if $\displaystyle f(a)=f(a')$ then $\displaystyle a=a'$. Start by choosing any $\displaystyle a,a'\in A$ such that $\displaystyle f(a)=f(a')$. Then, applying $g$ to both sides of $\displaystyle f(a)=f(a')$ shows that $\displaystyle g\circ f(a)=g\circ f(a')$. This function is injective and thus $\displaystyle a=a'$. This shows that $f$ is injective completing the proof.

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