Please show the work
This extra credit assignment comprising three problems asks you to explore and ultimately produce and prove an example of this phenomenon. I will not grade or assign nonzero credit to any submissions that do not contain a fully correct and complete answer to his part, which is question 7 below. You have one submission only.
In this assignment, A, B and C represent sets, g is a function from A to B, and f is a function from B to C, and h stands for f composed with g, which goes from A to C.
1. Prove that if the first stage of this pipeline, g, fails to be 1-1, then the entire pipeline, h can also not be 1-1. You can prove this directly or contrapositively.
2. Prove that if the second stage of the pipeline, f, fails to be onto, then the entire pipeline, h, can also not be onto.
3. Formulate the consequence of theorems 1. and 2. for a bijective h.
4. Prove that if g is not onto, and h is bijective, then f cannot be 1-1.
5. Prove that if f is not 1-1, and h is bijective, then g cannot be onto.
6. Explain what 3-5 implies for our search for finding non-bijective f and g with a bijective composition.
7. Inspired by your finding in 6, find three sets A,B,C, not necessarily different from each other, and two functions f and g so that g is a function from A to B, f is a function from B to C, and h is bijective, while neither f nor g is bijective. Prove that your example is correct.
8. Give a brief, intuitive explanation of how the specific ways in which your f and g fail to be bijective cancel each other out in a sense and make it possible for h to be bijective.