### Definition 8.2.1.

Let \(R\) be a relation on \(A\text{.}\) Then

- \(R\) is reflexive if for all \(x\in A\text{,}\) \(x R x\text{.}\) In ordered pair notation, \((x, x)\in R\text{.}\)
- \(R\) is symmetric if for all \(x, y\in A\text{,}\) if \(x R y\) then \(y R x\text{.}\) In ordered pair notation, if \((x, y)\in R\) then \((y, x)\in R\text{.}\)
- \(R\) is transitive if for all \(x, y, z\in A\text{,}\) if \(x R y\) and \(y R z\) then \(x R z\text{.}\) In ordered pair notation, if \((x, y)\in R\) and \((y, z)\in R\) then \((x, z)\in R\text{.}\)