Marcinkiewicz–Zygmund inequality: Difference between revisions

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In [[mathematics]], the '''symmetric closure''' of a [[binary relation]] ''R'' on a [[Set (mathematics)|set]] ''X'' is the smallest [[symmetric relation]] on ''X'' that contains ''R''.
 
For example,  if ''X'' is a set of airports and ''xRy'' means "there is a direct flight from airport ''x'' to airport ''y''", then the symmetric closure of ''R'' is the relation "there is a direct flight either from ''x'' to ''y'' or from ''y'' to ''x''". Or, if ''X'' is the set of humans (alive or dead) and ''R'' is the relation 'parent of', then the symmetric closure of ''R'' is the relation "''x'' is a parent or a child of ''y''".
 
== Definition ==
 
The symmetric closure ''S'' of a relation ''R'' on a set ''X'' is given by
 
:<math>S = R \cup \left\{ (x, y) : (y, x) \in R \right\}. \, </math>
 
In other words, the symmetric closure of ''R'' is the union of ''R'' with its [[inverse relation]], ''R''<sup>&nbsp;-1</sup>.
 
== See also ==
 
* [[Transitive closure]]
* [[Reflexive closure]]
 
== References ==
* Franz Baader and Tobias Nipkow, ''Term Rewriting and All That'', Cambridge University Press, 1998, p. 8
 
[[Category:Mathematical relations]]
[[Category:Closure operators]]
[[Category:Rewriting systems]]

Latest revision as of 10:14, 17 September 2013

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R.

For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the symmetric closure of R is the relation "there is a direct flight either from x to y or from y to x". Or, if X is the set of humans (alive or dead) and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parent or a child of y".

Definition

The symmetric closure S of a relation R on a set X is given by

S=R∪{(x,y):(y,x)∈R}.

In other words, the symmetric closure of R is the union of R with its inverse relation, R -1.

See also

References

  • Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8
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