Function and Non-Function

Relations Between Sets

In mathematics, a relation from a set $A$ to a set $B$ is a rule that connects members of set $A$ with members of set $B$. This pairing can be in any form.

Example:

The "less than" relation between $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4\}$ yields the pairs $(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)$.

Explanation:

We look for all pairs $(a, b)$ with $a \in A$ and $b \in B$ where $a < b$.

• For $a=1$ -> $1 < 2$, $1 < 3$, $1 < 4$. Pairs: $(1, 2), (1, 3), (1, 4)$.
• For $a=2$ -> $2 < 3$, $2 < 4$. Pairs: $(2, 3), (2, 4)$.
• For $a=3$ -> $3 < 4$. Pair: $(3, 4)$.

The combination of all these pairs is $\{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}$.

Functions as Special Relations

A function (or mapping) $f$ from a set $A$ to a set $B$, written $f: A \to B$, is a special relation that satisfies two conditions:

1. Every element $x \in A$ must have a pair $y \in B$.

   $$\forall x \in A, \exists y \in B \text{ such that } (x, y) \in f$$

2. Every element $x \in A$ has exactly one pair $y \in B$.

   $$\text{If } (x, y_1) \in f \text{ and } (x, y_2) \in f, \text{ then } y_1 = y_2$$

This means every member of the domain must be connected, and cannot have more than one connection.

Arrow Diagram Examples

Here are visual examples of relations using arrow diagrams to distinguish between functions and non-functions.

Relations That Are Not Functions

Relations That Are Functions