Improve Results for Set Identities

In this work, we presented a new law which was based on the well-known duality property for the set identities. We introduced the diagrams that could be applied to the proof for the set identities. Some prime examples were also provided to illustrate the proposed law.


Introduction
The set identities are used in many fields, including probability, statistics, propositional logic, Boolean algebra (the binary operations ∨, ∧ and a unary ˉ), lattices (the binary operations ∨ and ∧), BCL/BCL+-algebra [1]- [4] (the binary operation * ) and computer science.De Morgan's laws are a pair of transformation rules that are both valid rules of inference, and Venn diagrams are used to analyze logical arguments and to illustrate relationships between sets.
In the set theory and Boolean algebra, "Identities come in pairs" is now often stated as the relationship between the two identities in each pair we use the concept of a dual, it is certainly nothing new, such as the famous De Morgan laws, dating back to the 19th century, and expresses the duality of the set identities, but it has long been thought that the law of the double complement does not have duality.This is an interesting problem, shedding new light on the operations of sets.
Venn diagrams do not prove whether equality is true or not.So, this is the problem of basic theory, and we are to dope out a solution to the problem.
In this work we present a new law, and we also put the definition of difference evolved into law (i.e. the dual of difference-set law), domination laws and the identity laws.The results show that the diagram can meet proof of sets that this research could be useful to understand the set essence.

Set Identities
Table 1 presents some set identities that arise firstly.In general, we have the following law, and we prove that first.
( )  .Note that the so called "Liu's law" for propositions, named after the Yonghong Liu.Proof.We have The proof is completed.
The Liu's law is to fill gaps in an existing textbook [5].The most important set identities, such as domination laws and De Morgan's laws (first De Morgan's law and second De Morgan's law) are dual laws.

Diagrams and Definitions
As we know, Venn diagrams are often used to indicate the relationships among sets.But, it lacks the proof for set identities.New, here's the problem that we confront, and we must change and find new solutions.
Set can be represented graphically using curve.We draw a curved surface to indicate the universal set U , which is the set of the curve.We use notation S to denote the curve diagram of the sets, and S is a finite set.In curved surface, let  and  be a coordinate set of operation.Let       , , , ⊆      .We use the coordinate sets to show that ( ) ( )

Applications
The associative law is proved in Figures 9(a ( ) ( ) ( ) ( )  and ( ) ( ) ( ) ( ) Since the curve diagrams for ( ) and ( ) That is, the two curved surfaces are set-preserving equivalent.The identity of associative law is valid.

( ) ( ) ( )
  be a machine on the set.Let A , B , and C be sets with , , A B C S ⊆ .The  ( ) This is what we needed to prove.Example 4.1 Use set identities to prove that ( ) ( ) .
x A B B A B x x Whether the system has a unique solution be equivalent to saying that two equations .Y. H. Liu be a operation, then we have the following definitions.Definition 3.1 Let A and B be sets.Set B is a subset of set A , and is denoted by B A ⊆ .This is illustrated in Figure 1.That is, if and only if A B A =  , then B A ⊆ .Definition 3.2 The intersection of the sets A and B , denoted by A B  .This is illustrated in Figure 2. Definition 3.3 The union of the sets A and B , denoted by A B  .This is illustrated in Figure 3. Definition 3.4 The Liu's law is illustrated in Figure 4.

Figure 4 Definition 3 . 5
Figure 4.A B  .Definition 3.5 The difference of the sets A and B , denoted by A B − .This is illustrated in Figures 5(a) and (b).Definition 3.6 The complement of the set A , denoted by A .This is illustrated in Figures 6(a) and (b).Definition 3.7 The domination laws are illustrated in Figure 7. Definition 3.8 The domination laws can be decomposed into null set Φ and universal set U .This is illu- strated in Figures 8(a) and (b).

Law 4 . 1 (
Associative law).Use diagram to prove that a machine on the set.Let A , B , and C be sets with , ,

Figure 9
Figure 9 displays the graphs of the set ℘. Here, if and only if,

Figure 6 .
Figure 6.A .(a) The    model for A ; (b) The    model for A .

Figure 7 .
Figure 7. Set of the domination laws.

Law 4 . 2 (
Distributive law).Use diagram to prove that Figure 8. Φ and U . (a) The ith term of Φ ; (b) The ith term of U .