Functions:Bijective

A function is bijective if it is both injective and surjective. That is, a bijective function maps each element of the domain to a distinct element in the codomain, and every element in the codomain is mapped to by exactly one element of the domain.

Examples

Injective, not surjective:

• $f:A\to B,A=\{a,b,c\},B=\{1,2,3,4,5\}$ such that $f(a)=1$, $f(b)=2$, and $f(c)=3$ (each input has a unique output, but not all elements of the codomain are mapped to)
• $f:\mathbb {N} \to \mathbb {N} ,f(n)=n^{2}$
• $f:\{x\in \mathbb {R} ,x\geq 0\}\to \mathbb {R} ,f(x)=x^{2}+100$

Surjective, not injective:

• $f:A\to B,A=\{a,b,c,d\},B=\{1,2,3\}$ such that $f(a)=1$, $f(b)=1$, $f(c)=2$, and $f(d)=3$
• $f:\mathbb {R} \to \{x\in \mathbb {R} ,x\geq 0\},f(n)=|x|$
• $f:\mathbb {R} \to \mathbb {R} ,f(x)=x^{2}+2x+1$ ($-2$ and $0$ map to the same output, so not injective; range is $(-\infty ,\infty )=\mathbb {R}$, so surjective)

Bijections:

• $f:A\to B,A=\{a,b,c\},B=\{1,2,3\}$ such that $f(a)=1$, $f(b)=2$, and $f(c)=3$
• $f:\mathbb {R} \to \mathbb {R} ,f(x)=3x+5$
• $f:\mathbb {R} \to \mathbb {R} ,f(x)=x^{3}$

Resources

• Course Textbook, pages 154-164