# Definition:Fiber of Truth

(Redirected from Definition:Solution Set)

## Definition

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

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

$\set {x \in X: \map P x = \mathrm T}$

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

## 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.

### Solution

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

Let $S = \set {x \in X: \map P x = \mathrm T}$ be the fiber of truth (under $P$).

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

## 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.