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

Let : and be functions so that is an injection. Prove or disprove that the composition 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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Check out part (b) and it might give you a hint for this problem. Keep in mind also that we can use the proof from part (a) to show that if is injective, then must be injective. Thus, perhaps part (b) can help in this regard in finding a counter example. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We use a similar counterexample as in part (b) Let , , be the function sending 1 to 2 and let be the function sending all of B to 1. Then is the function sending 1 to itself, and hence injective. However is not injective since it sends both 2 and 3 to the element 2. This completes the counterexample. 