Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. We use the notation P(x) to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a collection of all the objects for which P is sensible and true.
For instance, {x | x is a positive integer less than 4} is the set {1,2,3}.
Thus, an element of {x | P(x)} is an object t for which the statement P(t) is true. Such a sentence P(x) is called a Predicate. P(x) is also called a propositional function, because each choice of x produces a proposition P(x) that is either true or false.
In formal semantics a predicate is an expression of the semantic type of sets. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory An equivalent formulation is that they are thought of as indicator functions of sets, i. In Mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of e. functions from an entity to a truth value. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function An entity is something that has a distinct separate Existence, though it need not be a material existence In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true
In first-order logic, a predicate can take the role as either a property or a relation between entities. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science In modern Philosophy, Mathematics, and Logic, a property is an Attribute of an object; thus a red object is said to have the property
See also
- Set-builder notation makes use of predicates
External links
In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation"Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic