Predicate

A predicate is a logical expression. In the context of set theory, usually a predicate is a statement which can be expressed using only symbols from symbolic logic, variables, and the set-theoretic relations $\in$ and $=$ . Additional relations can be made with predicates by using propositional logic quantifiers such as $\forall$ and $\exists$ , which mean "for-all" and "there exists" respectively.

Examples

$\varnothing \in y$

In English, this predicate reads, "The empty set is an element of $y$ ." Note that this is not true for all sets.

$\forall x, \varnothing \subseteq x$

In English, this translates to, "For all sets $x$ , the empty set is a subset of $x$ ." Since $A \subseteq B$ is an abbreviation for the predicate $\forall y (y \in A) \implies (y\in B)$ , this can be rewritten using only logical symbols, variables, and the set-theoretic notations $\in$ and $=$ , as follows:

$\forall x \forall y, (y \in \varnothing) \implies (y \in x)$ .

In English, this revised predicate reads, "For all sets $x$ , for all sets $y$ , if $y$ is an element of the empty set, then $y$ is an element of $x$ ."

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Predicate

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