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