# Relation/Examples/Ordering on Arbitrary Sets of Integers/Not Many-to-One

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## Example of Relation which is not Many-to-One

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 $\mathcal R$ 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 $\mathcal R$

- $\begin{array}{r|rrrr} A \times B & 1 & 2 & 3 & 4 \\ \hline 1 & \bullet & \bullet & \bullet & \circ \\ 2 & \bullet & \bullet & \circ & \circ \\ 3 & \bullet & \circ & \circ & \circ \\ \end{array}$

This relation $\mathcal R$ can be described as:

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

$\mathcal R$ is not a many-to-one relation.

## Proof

For example we have:

- $\tuple {1, 1} \in \mathcal R$

and:

- $\tuple {1, 2} \in \mathcal R$

Hence $\mathcal R$ is not many-to-one by definition.

$\blacksquare$

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Example $4.4$