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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