# Relation/Examples

## Examples of Relations

### Subsets of Initial Segment of Natural Numbers

Let $S$ be the set of all the subsets of the initial segment of the natural numbers $\set {1, 2, 3, \ldots, n}$.

Let $\RR$ be the set defined as:

$\RR = \set {\tuple {S_1, S_2}: S_1 \subseteq S_2, S_1 \in S, S_2 \in S}$

Then $\RR$ is a relation on $S$.

### Ordering on Arbitrary Sets of Integers

Let $A = \set {1, 2, 3, 4}$ and $B = \set {1, 2, 3}$ be sets of integers.

Consider the following diagram, where:

$A$ runs along the top
$B$ runs down the left hand side
a relation $\RR$ between $A$ and $B$ is indicated by marking with $\bullet$ every ordered pair $\tuple {a, b} \in A \times B$ which is in the truth set of $\RR$
$\begin {array} {r|rrrr} A \times B & 1 & 2 & 3 & 4 \\ \hline 1 & \bullet & \bullet & \bullet & \circ \\ 2 & \bullet & \bullet & \circ & \circ \\ 3 & \bullet & \circ & \circ & \circ \\ \end {array}$

This relation $\RR$ can be described as:

$\RR = \set {\tuple {x, y} \in A \times B: x + y \le 4}$

### Sisterhood

Let $S$ be the set of all human females.

Let $T$ be the set of all human beings.

Let $\RR$ be the set defined as:

$\RR = \set {\tuple {a, b}: a \in S, b \in T, \text {$a$is a sister of$b$} }$

Then $\RR$ is a relation in $S$ to $T$.