Mathematical logic and foundations Definition

Uniqueness quantifier

Builds on Existential quantifier Universal quantifier

Definition

Let $X$ be a set and let $P(x)$ be a predicate on $X$ . The proposition $\exists! x \in X, \, P(x)$ is true when exactly one $a \in X$ makes $P(a)$ true. In terms of the other quantifiers, it is the proposition $\exists x \in X, \, \bigl( P(x) \land \forall y \in X, \, (P(y) \to y = x) \bigr).$

Remarks

A proof of $\exists! x, \, P(x)$ usually has two parts: existence, an $x$ with $P(x)$ , and uniqueness, if $P(x)$ and $P(y)$ then $x = y$ . For example, $\exists! x \in \mathbb{R}, \, 2x + 1 = 7$ is true, with $x = 3$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Proposition concept
Predicate definition
Existential quantifier definition
Universal quantifier definition
Uniqueness quantifier definition

Where this leads

836 pages build on this one.

Tag	`0N3ADS4CHH`
URI	`m2://0N3ADS4CHH`
Kind	definition
Writing	W1 stated
MSC	03B10
Defines	Uniqueness quantifier
Uses	Existential quantifier, Universal quantifier

Uniqueness quantifier

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by