Extension (predicate logic)

Template:Short description The extension of a predicateTemplate:Spaced ndasha truth-valued functionTemplate:Spaced ndashis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.

Examples

For example, the statement "d2 follows the weekday d1" can be seen as a truth function associating to each tuple (d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuples associated with the value true, i.e.

{(Monday, Sunday),
 (Tuesday, Monday),
 (Wednesday, Tuesday),
 (Thursday, Wednesday),
 (Friday, Thursday),
 (Saturday, Friday),
 (Sunday, Saturday)}

By examining this extension, we can conclude that "Tuesday follows the weekday Saturday" (for example) is false.

Using set-builder notation, the extension of the n-ary predicate $\mathrm{\Phi }$ can be written as

$\{(x_1,...,x_n)\mid \mathrm{\Phi }(x_1,...,x_n)\}.$

Relationship with characteristic function

If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectivelyTemplate:Spaced ndashmaking the characteristic function a predicateTemplate:Spaced ndash, then for all relations R and predicates $\mathrm{\Phi }$ the following two statements are equivalent:

• $\mathrm{\Phi }$ is the characteristic function of R
• R is the extension of $\mathrm{\Phi }$

See also

• Extensional logic
• Extensional set
• Extensionality
• Intension

References

• extension (semantics) in nLab

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