Discrete mathematics-The concept of two-yuan relationship and closure

two USD relationshipset S is a non-empty set, and R is a condition of the elements of S. If any of the ordered element pairs (A, b) in S, we can always determine if a and B. satisfies the condition R, it is called R is a relationship of S (relation). If A and B satisfy the condition R, then A and B meet the condition R, It is said that A and B have a relationship with R, do the ARB; otherwise A and B have no relationship R. Relationship R also becomes two yuan relationship.
Definition:
The two-element relationship between set X and set Y is r= (x, Y, G (R)), which is the G (r), called R, and is a subset of the Cartesian product Xxy. if (x, y) ∈g (R) is said X is R-relation to Y and is recorded as XRy or R (x, y).

But often we equate relationships with their graphs, even if R? Xxy R is a relationship.

Closed Package

A closure of a relationship R is a new ordered even set with a reflexive, symmetric, or transitive form, coupled with a minimum number of ordered pairs, which is the closure of the relationship R. Set R is a two-element relationship on set a, R's reflexive (symmetric, transitive) closure is a relationship that satisfies the following conditions R ': (i) R ' is reflexive (symmetric, transitive); (ii) R '? R; (iii) for any reflexive (symmetric, transitive) relation on a R ", if R"? R, then there is R "? R '. The reflexive, symmetric, transitive closures of R are recorded as R (R), S (r), and T (r), respectively. Property 1 A two-tuple relationship on a set a closure operation of R can be compounded, for example: TS (r) =t (s (r)) represents the transitive closure of the symmetric closure of R, usually referred to as the symmetric transitive closure of R. The TSR (R) represents the self-rejecting transitive closure of R. Property 2 Set R is a two-dollar relationship on set a, there is (a) if R is reflexive, then S (r) and T (R) are also reflexive, (b) if R is symmetric, then R (r) and T (r) are symmetrical, and (c) if R is passed, then R (R) is also passed. Property 3 Set R is a two-element relationship on set A, there is (a) RS (r) =SR (R), (b) RT (R) =tr (R), (c) TS (R), St (R).

Discrete mathematics-The concept of two-yuan relationship and closure