Fiber of Truth/Examples

Examples of Solution Sets

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