Reflexive closure

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

$$S\; =\; R\; \backslash cup\; \backslash \{(x,\; x)\; \backslash mid\; x\; \backslash in\; X\backslash \}$$

In plain English, the reflexive closure of $R$ is the union of $R$ with the identity relation on $X.$

As an example, if

$$X\; =\; \backslash \{1,\; 2,\; 3,\; 4\backslash \}$$

$$R\; =\; \backslash \{(1,1),\; (1,3),\; (2,2),\; (3,3),\; (4,4)\backslash \}$$

then the relation $R$ is already reflexive by itself, so it does not differ from its reflexive closure.

However, if any of the reflexive pairs in $R$ was absent, it would be inserted for the reflexive closure.

For example, if on the same set $X$

$$R\; =\; \backslash \{(1,1),\; (1,3),\; (2,2),\; (4,4)\backslash \}$$

then the reflexive closure is

$$S\; =\; R\; \backslash cup\; \backslash \{(x,x)\; \backslash mid\; x\; \backslash in\; X\backslash \}\; =\; \backslash \{(1,1),\; (1,3),\; (2,2),\; (3,3),\; (4,4)\backslash \}\; .$$

• {{annotated link|Symmetric closure}}
• {{annotated link|Transitive closure}}

{{reflist}}

• Franz Baader and Tobias Nipkow, [https://books.google.com/books?id=N7BvXVUCQk8C&q=%22reflexive+closure%22 Term Rewriting and All That], Cambridge University Press, 1998, p. 8

Category:Binary relations

Category:Closure operators

Category:Rewriting systems

{{Order theory}}

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