The symmetric closure of a binary relation is , where means . Every symmetric relation containing contains all these reversed pairs, proving minimality.
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The symmetric closure of a binary relation \( R \) on a set \( A \) is a way to create a new relation that is symmetric while preserving the properties of the original relation. Specifically, a relation \( R \) is symmetric if for any elements \( a \) and \( b \) in set \( A \), if \( (a, b) \) is in \( R \), then \( (b, a) \) is also in \( R \).