# Definition:Precisely One Function

Jump to navigation
Jump to search

## Definition

Let $p_1, p_2, \ldots, p_n$ be statements.

The **precisely one function** is the propositional function $\map P {p_1, p_2, \ldots, p_n}$ defined as:

- $\map P {p_1, p_2, \ldots, p_n}$ is true if and only if
**precisely one**of $p_1, p_2, \ldots, p_n$ is true.

This page needs the help of a knowledgeable authority.In particular: It's got a name but I can't find it anywhere. I've seen it (we had something like this up on $\mathsf{Pr} \infty \mathsf{fWiki}$ a decade ago but we must have deleted it for some reason). Anyone?If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: Put the below into its own page as a derived resultYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

- $\exists p: p \in \set {p_1, p_2, \ldots, p_n}: \paren {p = \text {True} } \implies \forall x_1, x_2 \paren {x_1, x_2 \in \set {p_1, p_2, \ldots, p_n}: \paren {x_1 = \text {True} \land x_2 = \text {True} \implies \paren {x_1 = p} \land \paren {x_2 = p} } }$

## Also see

- Results about
**the precisely one function**can be found here.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(6)$