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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]


  1. [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]


  1. 1.0 1.1 Types and Programming Languages - Benjamin C. Peirce