What is closure of relation in discrete mathematics?

Answer 1

In general, the closure of some relation R is only defined with respect to some property P. Given a property P and a relation R, the closure of R is defined by including the minimum number of ordered pairs to R such that R satisfies P.

For example, take the set S = {a,b,c} and consider the relation R on S x S given by R = {(a,c),(b,b),(c,b)}. A relation has the property of being symmetric if for all x,y, xRy implies yRx. Thus the symmetric closure of R is {(a,c),(b,b),(c,b),(c,a),(b,c)}.