# 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 $=$.

## 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 $x$."