Relations
Operators such as = and <, which compare two elements of the same kind and yield a truth value as a result, are called relations. A relation ~ on elements of a set U can be considered as the set {(u,v)∈U×U : u~u} of pairs of elements that satisfy the relation. Alternatively, ~ can be considered as a predicate with two variables, each of which has U as its domain.
The Reflexive, Symmetric, and Transitive Properties
The definition of > in the earlier rock-scissors-paper example wasn’t typical. Most directional or bidirectional comparison operators in mathematics, such as <, ≥, and =, are transitive. Here’s a precise definition of the transitive property and some other useful properties a relation can have.
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