Definition:Membership Relation

Definition

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

Let $\RR \subseteq S \times \powerset S$ be the relation defined as:

$\tuple {x, A} \in \RR \iff x \in A$

Thus $\RR$ is the relation between elements of $S$ and subsets of $S$ expressing membership.

Also see

• Results about the membership relation can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
• 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Introduction