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Relation
A relation describes as a connection between the elements of a set, and that may or may not hold between two elements of that set.
For example, in the set of natural numbers, the relation "is less than" does not hold between 5 and 4, but it does between 4 and 5.
Formal Definition
Given a set \(X\), a relation \(R\) over \(X\) is a set of ordered pairs of elements from \(X\), then:
\(R \subseteq \{(x,y)|x,y \in X\}\)
The statement \((x,y) \in R\) reads "\(x\) is \(R\)-related to \(y\)", and is written in the infix notation as \(xRy\). The order of elements is important, because if \(x \neq y\), then \(yRx\) can be true or false independently of \(xRy\).
Related
- Finitary Relation (a relation on the Cartesian product of a number of sets)
- Binary Relation (a widely studied finitary relation in which the number of sets is 2)