Examples on Injective, Surjective, and Bijective functions

Example 12.4.

Proposition: The function \(f:\mathbb{R}-\{0\}\to\mathbb{R}\) defined by the formula \(f(x)=\frac{1}{x}+1\) is injective but not surjective.

Example 12.5.

Proposition: The function \(f:\mathbb{R}-\{0\}\to\mathbb{R}-\{1\}\) is injective and surjective (hence bijective).

Example 12.6

Proposition: The function \(g:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}\) defined by the formula \(g(m,n) = (m+n,m+2n)\) is both injective and surjective.

Example 12.15

Let \(A=\{A,B,C,D,E,F,G\}\) and let \(B=\{1,2,3,4,5,6,7\}\). How many functions are there from \(A\) to \(B\)? How many of these are injective? How many are surjective? How many are bijective?