Example: Examples of Functions in a Logical Calculus

(related to Definition: Function, Arity and Constant)

Example 1

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

$s$: "There exists a natural number $x$ such that the equation $x+2=5$ holds."

In this string, $2$ and $5$ are constants and $x+2=5$ defines a binary function since it takes two arguments - the variable $x$ and the constant $2$ as input and maps these arguments to the constant $5$ as output.

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, the substring \("0"\) is a constant. The substrings \("f"\) and the absolute value \("|\cdot|"\) are unary functions, because they take one argument as input. The strings containing the subtraction sings \(" x - a"\) and $f(x)-f(a)$ are binary functions, because they take two arguments and assign to them impolicitely a real value.

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