Example: Examples of Predicates in a Logical Calculus

(related to Definition: Predicate of a Logical Calculus)

Example 1

Let $s$ be the string "$x\in\mathbb N: x+2=5$", let $U$ be the domain of discourse of natural numbers $\mathbb N:=\{0,1,2,\ldots,\}$ and let $I(s)$ be the interpretation of $s$ assigning it a meaning in $U$. Then the string $s$ has the meaning:

$s$: "an element $x$ of the set of natural numbers $\mathbb N$ such that $x+2=5$"

In this string, "$x\in N$" is a unary predicate, since it takes only one argument $x$ as input.

Example 2

Take real numbers as the domain of discourse, and consider the \(\epsilon-\delta\) definition of continuous real functions:

A real function \(f:D\to\mathbb R\) is continuous at the point \(a\in D\), if for every \(\epsilon > 0\) there is a \(\delta > 0\) such that \(|f(x)-f(a)| < \epsilon\) for all \(x\in D\) with \(|x-a| < \delta.\)

This proposition can be codified using a string like this:

"$\forall\epsilon\,(\epsilon > 0)\,\exists\delta\,(\delta > 0)\,\forall x\,(x\in D)\,(|x-a|<\delta\Longrightarrow|f(x)-f(a)|<\epsilon).$"

In this string, \("x \in D"\) is a unary predicate, because it takes one argument \(x\) to indicate that is contained in \(D\). The comparison relation \( " < " \) is a binary predicate because it compares two arguments.

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