Prove or disprove the following statement: if f is an injective function from A to B, and g is a surjective function from B to C, then gof is a bijective function from A to C.
Really need help. I'm so lost.
You might find Chapter 3 of Part I of Topology and the Language of Mathematics useful. It provides a number of example of problems like yours, with solutions.
A free download of the first 50 pages (which includes Chapter 3 of Part I) is available here:
Bobo Strategy - Topology
Hope it's useful -
Chris
Making the second of your maps the constant map while the first is a non-surjective injection will provide quite a neat solution.
For instance, , . Then, taking the constant map, , .
Here, the composition of and is just the first map, \phi. Clearly it is injective but not surjective.