In this recitation, we went through textbook exercises 4.6, 4.10, 4.11, 4.22, and 4.25 parts d and e. The point here is to get students be familiar with the technique for proving bijectivity, injectivity and surjectivity.
- To prove the injectivity of a function, say f: A\to B, pick any two elements x_1, x_2\in A such that f(x_1)=f(x_2), and with some work show x_1=x_2.
- To prove the surjectivity, pick any y\in B, and show f(x)=y has solution(s).
- There are two ways to show that f is bijective. The first method is to show it is both injective and surjective. The second method is to show f(x)=y has a unique solution in A. Usually, the byproduct of the second approach is the inverse function of f.