Antitransitive Relation/Examples/Greater by One

Example of Antitransitive Relation

Let $\RR$ denote the relation on the set of real numbers $\R$ defined as:

$x \mathrel \RR y \iff \text {$x$ is greater than $y$ by $1$}$

that is:

$x - y = 1$

Then $\RR$ is an antitransitive relation..

Proof

Let $x \mathrel \RR y$ and $y \mathrel \RR z$.

Then:

\(\text {(1)}: \quad\)

\(\ds x - y\)

\(=\)

\(\ds 1\)

Definition of $\RR$

\(\text {(2)}: \quad\)

\(\ds y - z\)

\(=\)

\(\ds 1\)

Definition of $\RR$

\(\ds \leadsto \ \ \)

\(\ds x - z\)

\(=\)

\(\ds 2\)

adding $(1)$ and $(2)$

Hence by definition:

$\mathop \neg x \mathrel \RR z$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): transitive relation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): transitive relation