Recitation 13

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 $f$.