Sunday, December 17

Set Theory And Set Theory Algebra

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Introduction

In this article I will discuss the basic concepts of set theory and the algebra of set theory. As well, I will highlight how it can be useful in number theory and to study relation between different sets. As well I will also use venn diagrams to illustrate its usefulness in probability problems and in intersection and union of sets.

Basic definitions in set theory

The basic definition of set theory relates to the definition of a set, element of a set, finite set, infinite set, set equality, subset, complemetary set, universal set, empty set, intter section and union of sets, relative complements.

Definition of a set

A set can be defined by describing the properties of a set or by listing all the lements separated by commars. For example a set cab be defined as below:

A = { 1, 2, 3, 4 , 5 , 6} or it can be defined as A = x:: x is all integers between 0< x < 8)

Element of a set

The lement of a set is a particualr member of a set say “a” a member of a set is in the set then “a ” is an element of a set. It is mostly described as a ∈ X.

Finite set

A set where the elements of a set can be listed and the elements are limited and not infinite. For example, as et of numbers which are even betweeb 1 and 50.

Infinite sets.

In numbers most of the natural numbers, fractions are infinite sets because the lements are infinfite and there is no lomit to the elements. For example a set of all fractions is an infinite set.

Set equality

If a set A has the same number of elements compared to set B and they are identical and the order not necessarily the same are called equal sets. That is if the above condition is applicable then, set A = set B.

Sub-set and proper subset

 A set can be a sub-set of another set if the elements of the former is also elements of the latter set. However, if any of the latter is not the elemets of the former then the former is a proper sub-set of latter set. Normally sub-set uses the following notation:

B ⊂ A.

For any two sets A and B if A is a sub set B and B is a subset of a then A = B. In addtion all sets are subset of itself.

Empty set or Null sets

If ther is no elemets and the number of lements is zwro is called a emty set. All empty sets are subsets of all sets.

The empty set is denoted by the following notation:

Emty or null set = Ø. Examples of empty sets are numbers which are even and odd. Cars with 50 doors, a set of squares with 5 sides.

Complementary set

Complement of a set is all the elments which are not the lements of A in the universal set. For example say the universal set is { 1, 2 3, 4, 5, 6, 7, 8, 9, 10, 11 , 12, 13 , 14  15, 16, 17 , 18 18, 20} if say the A is the even numbers of the universal set. Then the complemetary set is the odd numbers of the universal set.  That is the number of the even set + the number of complementary set = the number of elements in the universal set.  The complemetary set is denoted by A’

Universal set

Universal set is a set which is under consideration of study. For example it can be prime numbers between 1 and 1000 and one wants to study the prime numbers between 1 and 500 to that of 5001 to 1000.

The union of sets

The union of sets is the set of all elemets of sets concerned but not counting the common lements of the sets concerned. Say a set A = { 3, 4, 7, 9, 11} and set B = { 2, 3, 4, 7, 8, 9 11, 12}. Then A union B is { 2, 3, 4, 7, 9, 11 12}. Not all the elments of A and B but eleminating the common elements. The union is represented by the following notation:

A U B

Intersection of sets.

Intersection of sets

This the set of common elements of all the sets concerned in the intersection. In the above example the the intersection set of A and B is = { 3, 4, 7, 9 11}.

Laws of algebra applicable to set theory

If set A, set B and set C has union and intersection, then the laws below is applicable to intersections and Unions. The specific laws are as follows:

Commutative Law

A  intersection B = B intersection A

A Union B = B Union A

Associative Law

( A union B) union C = A union ( B union C)

( A intersection B) intersection C = A intersection ( B intersection C)

A union ( B intersection C) =  ( A union B ) intersection ( A union C)

A intersection ( B union C) = ( A intersection B) union ( A intersection C)

For any subset A of universal set U and and empty set or null set the following laws are applicable

Identity laws

A union empty set = A

A intersection U = A

Complement laws

A union ( complement set A) = U

A intersection ( complement set A) = empty set or null set

Additional Laws of union and intersections which can be derived from the above laws are as follows:

Principle of duality laws

Idempotent laws

A union A = A

A intersection A = A

Domination law

A  union U = U

A intersection ( empty set) = empty set

Absorption laws

A Union ( A intersection B) = A

A intersection ( A union B) = A

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