Discrete Mathematics--a summary of two-yuan relationship

Equivalence relationship:

Set R is a two-dollar relationship on set a, if R satisfies://Both are arbitrary elementsReflexivity: ∀a∈a, = (A, a) ∈r symmetry: (A, b) ∈r∧a≠b = (b, a) ∈r transitivity: (A, B) ∈r, (b, c) ∈r = (A, c) ∈r "R" is an equivalence relationship defined on a. Set R is an equivalence relationship, if (A, B) ∈r, then A is equivalent to B, recorded as a ~ b.Partial order Relationship: the partial order exists a<b,a<C, the phenomenon that the size between b and C cannot be compared. and the corresponding whole order must be shaped like A<B<C form. That is, the whole order requires each element to compare size, the partial order is not required. now the partial-order symbol and the quasi-order symbol? Or, the above is the old version, in order to prevent confusion. Set R is set AOn a two-dollar relationship, if R satisfies://Both are arbitrary elementsⅠ reflexivity: to any x∈ AYes xR x; Ⅱ anti-symmetry (that is, opposing the relationship): to any x, y∈ AIf xR yAnd yR x, you x= y; Ⅲ transitivity: to any x, y, Z∈ AIf xR yAnd yR Z, you xR Z。[1]//With a condition that satisfies the transitivity, the former is FalseThe R is called AThe partial order relationship on the upper side is usually written as a. Notice here? It is not necessary to refer to "less than or equal" in the general sense. If there is x? y, we also say xRow in yFront xPrecedes y）。Basic RelationshipsReflexivity: ∀a∈a, = = (A, a) ∈r anti-reflexive: ∀a∈a, = (A, a) ∉r symmetry: (A, b) ∈r∧a≠b = (b, a) ∈r//objection: (A, B) ∈r∧ (b, a) ∈r =>a= b//These three note that the previous piece is false transitivity: (A, B) ∈r, (b, c) ∈r = (A, c) ∈r//closure of "relationship" (Closure)In discrete mathematics, a closure of a relation R, which is formed by adding the smallest number of ordered pairs reflexive Nature， symmetry ofor transitivityThe new ordered even set, which is the closure of the relationship R.

Discrete Mathematics--a summary of two-yuan relationship