Predicate

Tags

logic

Are propositions defined over (free) variables. For example in the set \( \{ x: x \text{has property} P \} \), the property \( P \) is a [see page 11, predicate] denoted as \( P(x) \).

The truth value of a predicate depends on one or more free-variable, such as \( x \) and specifying a value for \( x \) produces a proposition. We define the see page 12, universe of discourse as the set of possible values for a free variable. More formally it may be [see page 23, defined] as the set of entities over which certain variables of interest in some formal treatment may range. In general when given a problem statement we're told what the universe of discourse is. The truth set is the subset of the universe of discourse, the set of \( x \), for which \( P(x) \) is true.

Note: When a predicate has more than one free variable its universe of discourse is a cartesian product, I.E. elements of it are tuples and the predicate takes a tuple of values as its input.