Difference between revisions of "Predicate"
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| − | 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 <math>\in</math> and <math>=</math>. | + | 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 <math>\in</math> and <math>=</math>. Additional relations can be made with predicates by using propositional logic quantifiers such as <math>\forall</math> and <math>\exists</math>, which mean "for-all" and "there exists" respectively. |
== Examples == | == Examples == | ||
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Latest revision as of 17:07, 26 March 2022
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 
and 
. Additional relations can be made with predicates by using propositional logic quantifiers such as 
and 
, which mean "for-all" and "there exists" respectively.
Examples
In English, this predicate reads, "The empty set is an element of 
." Note that this is not true for all sets.
In English, this translates to, "For all sets 
, the empty set is a subset of ." Since 
is an abbreviation for the predicate 
, this can be rewritten using only logical symbols, variables, and the set-theoretic notations and , as follows:
.
.In English, this revised predicate reads, "For all sets , for all sets , if is an element of the empty set, then is an element of ."
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