Functions

A function is a mapping of a set to another which assigns an element of the domain to one element in the codomain.

Relations

Let $f$ be a function from set $A$ to set $B$, denoted $f:A\to B$.

• $A\Rightarrow$ domain
• $B\Rightarrow$ codomain
• $f(a)=b$, $b$ is the image of $a$ and $a$ is the pre-image of $b$.
• the range of $f$ is the set of all images of elements of $A$.
• Let $S$ be a subset of $A$; the image of $S$ is a subset of $B$ that consists of the image of the elements of $S$, s.t.

$f(S)=\{f(s)\mid s\in S\}$

The diagram shows $f:A\to B$ with $A=\{a_1,a_2,a_3,a_4\}$ mapping into $B=\{b_1,b_2,b_3,b_4\}$; the dashed mapping is annotated “Not allowed!” (an element cannot map to more than one image).

Injective function

$f:A\to B$: for every $a_1,a_2\in A$, $f(a_1)=f(a_2)\Rightarrow a_1=a_2$. Every element of the domain has a distinct assignment in the codomain.

Surjective function

For every $b\in B$, there is at least one element $a\in A$ s.t. $f(a)=b$.

Bijective function

Both injective and surjective.

Inverse function

Let $f:A\to B$ be a bijective function; the inverse function is denoted by:

$f^{-1}:B\to A$

For every $b\in B$, $f^{-1}(b)=a$, iff $f(a)=b$.