Spherical law of cosines: Difference between revisions

Revision as of 20:45, 28 July 2013

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

For example the statement "d2 is the weekday following 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 is the weekday following Saturday" (for example) is false.

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

\{(x_{1},...,x_{n})\mid \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 $\Phi$ the following two statements are equivalent:

$\Phi$ is the characteristic function of R;
R is the extension of $\Phi$ .

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+The '''extension''' of a [[predicate (mathematics)|predicate]]{{spaced ndash}}a [[truth-value]]d [[function (mathematics)|function]]{{spaced ndash}}is the [[Set (mathematics)|set]] of [[tuple]]s of values that, used as arguments, satisfy the predicate. Such a set of tuples is a [[relation (mathematics)|relation]].
+For example the statement "''d2'' is the weekday following ''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 is the weekday following Saturday" (for example) is false.
+Using [[set-builder notation]], the extension of the [[n-ary|''n''-ary]] predicate <math>\Phi</math> can be written as
+:<math>\{ (x_1,...,x_n) \mid \Phi(x_1,...,x_n) \}\,.</math>
+==Relationship with characteristic function==
+If the values 0 and 1 in the range of a [[Indicator function|characteristic function]] are identified with the values false and true, respectively{{spaced ndash}}making the characteristic function a predicate{{spaced ndash}}, then for all relations ''R'' and predicates <math>\Phi</math> the following two statements are equivalent:
+*<math>\Phi</math> is the characteristic function of ''R'';
+*''R'' is the extension of <math>\Phi</math>.
+== See also ==
+* [[Extensionality]]
+* [[Intension]]
+[[Category:Predicate logic]]
+{{logic-stub}}
+{{mathlogic-stub}}

Spherical law of cosines: Difference between revisions

Revision as of 20:45, 28 July 2013

Relationship with characteristic function

See also

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