Definition:Comparable Elements

Definition

Let $\RR$ be a relation.

Two elements $x \in \Dom \RR$, $y \in \Img \RR$ such that $x \ne y$ are comparable if either:

$x \mathrel \RR y$

or:

$y \mathrel \RR x$

The concept is usually encountered when $\RR$ is an ordering.

Also see

• If $x$ and $y$ are not comparable then they are non-comparable.

• Results about comparable elements can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering