Definition:Variable/Satisfaction

Definition

Let $\map P x$ be a propositional function such that $x$ is a variable with a given domain $S$.

Let a specific element $a$ of $S$ be substituted for $x$ in $\map P x$ such that $\map P a$ is true.

Then $a$ is said to satisfy the propositional function $P$.

Tuple

An ordered $n$-tuple $T$ is said to satisfy an open statement $S$ if and only if the predicate of $S$ is true of $T$.

Examples

"Father" Relation

Consider the open statement:

$x$ was the father of $y$.

This is satisfied by the ordered pair:

$\tuple {\text {Laertes}, \text {Odysseus} }$

because Laertes was the father of Odysseus.

Also see

• Results about satisfaction can be found here.

Sources

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