Relations

Definition#

Let \(S\) and \(T\) be sets. Then a relation from \(S\) to \(T\) is a subset \(\rho\) of \(S \times T\) . If \(s \in S\) and \(t \in T\), then we write \(s \rho t\) if \((s,t) \in \rho\); otherwise, we write \(s \not\rho t\).

Informally: A relation is some kind of relationship between two objects in a set.

Example#

Let \(S = \{1,2,3\}\) and \(T = \{1,2,3,4\}\). Define a relation \(\rho\) from \(S\) to \(T\) via \(s \rho t\) if and only if \(st^2 \le 4\). Then \(\rho = \{(1,1), (1,2), (2,1), (3,1)\}\).

Source#

Gregory T. Lee. Abstract Algebra. 1st ed. Springer Cham, 2018. pp. 5.