Axiomatic of sets that we present is the one proposed by the Soviet mathematician, Gorbátov; it is the formal construction of set theory, using for that matter a system of adequate axioms and with the aim of using it in Computer Science topics. [1] . The name that is assumed is that of the author of the written work that appears mentioned in references on this page.
Summary
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- 1 Axiomatic
- 1 Axiom of existence
- 2 Axiom of bulkiness
- 3 Union Axiom
- 4 Axiom of difference
- 5 Power Axiom
- 6 Axiom of existence of the empty set
- 2 Other joint operations
- 1 Intersection
- 2 Complement
- 3 Symmetric difference
- 3 Properties of operations
- 4 References
- 5 Bibliographic source
- 6 See also
Axiomatic
Axiom of existence
There is at least one set. [2]
Bulk Axiom
Also called the extensionality axiom . If the sets M a and M b are made up of the same elements, they coincide (they are the same): M a = M b [3]
Union Axiom
For two arbitrary sets M a and M b there is a set, the elements of which are all the elements of the set M a and all the elements of the set M b and which contains no other elements.
From the axioms of voluminosity and of union it is inferred that for the arbitrary sets M a and M b the set that satisfies the axiom of union is unique. Uniqueness flows from the above. [4]
Difference Axiom
For arbitrary sets M a and M b there is a set, whose elements are those, and only those, elements of the set M a that are not elements of the set M b .
Similarly, it follows from the second and fourth axioms that for two arbitrary sets there exists exactly one set containing the elements of the first set, not belonging to the second. This set is called the difference of M a and M b and is denoted M c = M a – M b [5]
Power axiom
For each set M there is a family of sets B (M) (Boolean), whose elements are all subsets M i , M i ⊂ M, and only these. [6]
Axiom of existence of the empty set
There is such a set ∅ that no element belongs to it.
Other joint operations
Although the operations and concepts of set theory were intuitively elaborated, the axiom structuring gives the opportunity to formally define these concepts and operations of set theory, relying on the six axioms previously presented. With the help of union and difference, using the introduced axioms, we are going to define three more operations on sets. [7]
Intersection
It is defined by the formula M a In M b = M a – (M a – M b ) where In: = intersection,
It can be shown that the elements of the intersection A In B are those and only those that are elements of M a and are also elements of the set M b .
Complement
This operation is defined by the formula M c = 1 – M, where M is a proper part of 1 , the complement is different from Ø (empty set) and 1 , the universal set.
Symmetric difference
This operation is established using the defining formula (M a – M a ) U (M b – M a ).
The symmetric difference of two sets is the union of the two differences that can be determined with the given sets.
Operations Properties
Using the introduced axiomatic, the validity of the laws adduced to determine the properties of the algebra signature of sets can be tested: the “idempotence”, “commutative”, “associative”, “distributive” “laws of operation with constants »,« Double complement »,« laws formulated by De Morgan »and others presented below:
- distributive law of intersection with respect to difference
- commutative law of symmetric difference
- associative law of symmetric difference
- distributive law of intersection with respect to the symmetric difference
- queuing lawsthe set M can be obtained joining its intersections with N and the complement of this; otherwise, intersecting the union of M with N and with its complement.
M = (M In N) U (M In N c ) …. M = (MUN) In (MUN c )
- absorption laws
- Poretski’s laws. MUN = MU (N In M c ) … M In N = M In (NUM c )
