Predicate
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Definition
A predicate, [ilmath]P[/ilmath], is a [ilmath]1[/ilmath]place relation on a set [ilmath]X[/ilmath]^{[Note 1]}.
 We say "[ilmath]P[/ilmath] is true of [ilmath]x\in X[/ilmath]" if [ilmath]x\in P[/ilmath]^{[1]}
 We write [ilmath]P(x)\iff x\in P[/ilmath] to emphasise that [ilmath]x[/ilmath] has the predicate^{[1]}
See also
 Axiom of schema of comprehension  This states that given a set [ilmath]A[/ilmath] we can construct a set [ilmath]B[/ilmath] such that [ilmath]B=\{x\in A\ \vert P(x)\}[/ilmath] for some predicate [ilmath]P[/ilmath]
Notes
 ↑ [ilmath]P\subseteq X[/ilmath] in this case. In contrast to a binary relation[ilmath]\subseteq X\times X[/ilmath] or an [ilmath]n[/ilmath]place relation[ilmath]\subseteq \underbrace{X\times X\times\ldots\times X}_{n\ \text{times} } [/ilmath]
References
 ↑ ^{1.0} ^{1.1} Types and Programming Languages  Benjamin C. Peirce
