Logical Predicate

A Mathematical Statement $P$ about a variable $x$.

These are translations from English sentences to logical statements that take an arbitrary Free Variable like P(x). The quantifiers can be evaluated with an input to return true or false They tend to paired with Logical Connectives and Logical Predicate

Definitions

Function Definition

With a non-empty Domain: A predicate is a function $A:D^n\to \{0,1\}$ with Arity $n$

1 \text{ if } (x_{1},\dots,x_{n}) \in A\\ \\ 0 \text{ otherwise} \end{cases}$$ (Like an [[Indicator Random Variable|Indicator RV]]) ### [[Relation]] Definition With a non-empty [[Domain]]: A predicate is the [[Relation]] $A \subset D^{n}$ $$A = \{ (x_{1},\dots,x_{n}) \in D_{n} | A(x_{1},\dots,x_{n})=1 \}$$ # Example Predicates $P(n) : n^{2} = n$ Assumed by default to apply to all $n$. Don't add any [[Quantifier|Quantifiers]] in the definition. # Types - [[For All|Universal Quantifier]] - [[There Exists|Existential Quantifier]] - [[Bound Quantifier]] # Laws - [[Quantifier Negation Laws]]