Relation/Examples
Jump to navigation
Jump to search
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 $T$.