# Definition:Fiber of Truth

## Definition

Let $P: X \to \set {\T, \F}$ be a propositional function defined on a domain $X$.

The **fiber of truth (under $P$)** is the preimage, or **fiber**, of $\T$ under $P$:

- $\map {P^{-1} } \T := \set {x \in X: \map P x = \T}$

That is, the elements of $X$ whose image under $P$ is $\T$.

### Solution

Let $P: X \to \set {\T, \F}$ be a propositional function defined on a domain $X$.

Let $S = \map {P^{-1} } \T$ be the **fiber of truth (under $P$)**.

Then an element of $S$ is known as a **solution** of $P$.

## Also known as

The **fiber of truth** is often referred to also as the **solution set** for $P$.

This is particularly the case in mathematical contexts.

Some sources denote the **fiber of truth under $P$** as $\sqbrk {\size P}$.

## Also see

## Examples

### Solution Set of $x^2 = 2$ in $\R$

Let $x$ denote a variable whose domain is the set of real numbers $\R$.

Let $\map P x$ be the propositional function defined as:

- $\map P x := x^2 - 2$

Then the solution set of $\map P x$ is $\set {\sqrt 2, -\sqrt 2}$.

### Solution Set of $x^2 = 2$ in $\Q$

Let $x$ denote a variable whose domain is the set of real numbers $\Q$.

Let $\map P x$ be the propositional function defined as:

- $\map P x := x^2 - 2$

Then the solution set of $\map P x$ is the empty set $\O$.

### Solution to $x^2 - 2 x - 3$

Consider the equation in algebra:

- $x^2 - 2 x - 3 = 0$

where the domain of $x$ is implicitly taken to be the set of real numbers $\R$.

Then $3$ is a solution to $x^2 - 2 x - 3 = 0$.

## Linguistic Note

The phrase **fiber of truth** (with the same meaning) is occasionally seen in natural language.

In particular:

*... to extract the fiber of truth from this tissue of lies ...*

sounds as though it would be used in the context of the courtroom by a lawyer waxing rhetorical.

The British English spelling of **fiber** is **fibre**. The pronunciation is the same.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers