Appendix B: Relations and Orderings

Abstract

Let Ω be a non-empty set. A relation on Ω is simply a subset of the Cartesian square of Ω. In symbols we could write $R \subseteq \Omega \times \Omega$. Suppose that (a, b) ∈ R. A very suggestive alternative way to express this is to use infix notation and write aRb to mean (a, b) ∈ R. For some reason, probably habit, people seem to prefer some non-alphabetic symbol when using infix notation. Let us pander to this, and write a ~ b as yet another synonym for (a, b) ∈ R. If Ω ₁ is a non-empty subset of Ω then R∩ (Ω ₁ × Ω ₁) is the induced relation on Ω ₁, though it usually is still called R (the benign supposition being that the set is evident from the context).