"Predicate" vs. "Relation" What's the difference between a predicate and a relation? I read the definition that an $n$-ary predicate on a set $X$ is a function $X^n\to \{\text{true}, \text{false}\}$ where $\{\text{true}, \text{false}\}$ is the set of truth values. Also, it is well-known that an $n$-ary relation is  simply a subset of $X^n$. Trivially there is a canonical bijection between the set of predicates on  a particular fixed set and the set of all relations on that set. So why do these two terms "predicate" and "relation" exist? Because of historical reasons? Is there a crucial difference between them?

A predicate is a type of relation.


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*A relation is a particular type of set.

*A function is a particular type of relation.

*A predicate is a particular type of function.


The bijection between relations and predicates you have described means they are conceptually interchangeable in most applications. But definitionally, relation is prior to predicate.
