Every function is a relation, but not every relation is a function. A relation is just two sets — a domain and a range — tied together by a fixed rule. One rule decides which relations are functions.

Concept One relation, four ways to write it

Take the rule “Khalid is always two years older than Ahmed.” The same relation can be given in any of these forms.

Verbally — Khalid's age exceeds Ahmed's age by two years.

Numerically —

5 \to 7

\; 6 \to 8

\; 2 \to 4

Algebraically —

K = A + 2

Graphically —

Concept The arrow diagram

Put the domain on the left, the range on the right, and draw an arrow for each pair the rule connects. Here

1 \to 3

2 \to 4

3 \to 5

— the arrow-diagram version of

K = A + 2

Concept The one rule for a function

Every element of the domain must point to exactly one element of the range.

Two arrows landing on one range element is fine — $1 \to 6$ and $2 \to 6$ together are allowed. One domain element sending two arrows is not — $1 \to 2$ and $1 \to 4$ breaks the rule.

Example The pairs test

Given a set of pairs, check whether any first value appears twice with different partners.

\{(1,2),\ (2,4),\ (3,6),\ (4,8)\}

— every first value is used once. A function.

\{(1,5),\ (1,7)\}

— the input

1

has two different outputs. Not a function.

\{(2,3),\ (4,3),\ (6,3)\}

— domain

\{2,4,6\}

, range

\{3\}

. No input repeats, so still a function.

Summary

A relation is a domain and a range linked by a fixed rule.
The same relation can be written verbally, numerically, graphically, or algebraically.
A function: every domain element points to exactly one range element.
Many-to-one is allowed ( $1 \to 6$ , $2 \to 6$ ); one-to-many is not ( $1 \to 2$ and $1 \to 4$ ).
In a pair list, a repeated first value with different second values means it is not a function.

Relation vs Function