# Definition:Precisely One Function

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## 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.

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

## 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 $(6)$