Simplified Overview
Informally, a predicate is a statement that may be true or false depending on the values of its variables. It can be thought of as an operator or function that returns a value that is either true or false. For example, predicates are sometimes used to indicate set membership: when talking about sets, it is sometimes inconvenient or impossible to describe a set by listing all of its elements. Thus, a predicate P(x) will be true or false, depending on whether x belongs to a set.
Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) is written as {x | P(x)}, and is just a collection of all the objects for which P is true.
For instance, {x | x is a positive integer less than 4} is the set {1,2,3}.
If t is an element of the set {x | P(x)}, then the statement P(t) is true.
Here, P(x) is referred to as the predicate, and x the subject of the proposition. Sometimes, P(x) is also called a propositional function, as each choice of x produces a proposition.
Read more about this topic: Predicate (mathematical Logic)
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