Set Theory

Set theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. Although a simple idea, it is, nevertheless, one of the most important and fundamental concepts in modern mathematics.

 

A set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. As can be seen from this example, sets are allowed to have an infinite number of elements.

 

If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x ∈ A. (The symbol "∈" is a derivation from the Greek letter epsilon, "ε"). The symbol  is sometimes used to write x ∉ A, or "x is not in A".

 

Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A. Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If A and B are equal, then this is denoted symbolically as A = B (as usual).

 

An empty set, often denoted by "{}" is a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. A Universal Set is a set consisting of all objects or elements of a type or of a given interest and is normally depicted by the alphabets X, U or S.  A Finite Set is one in which the number of elements can be counted.  An Infinite Set is one in which the number of elements cannot be counted. 

 

Given two sets A and B we say that A is a subset of B if every element of A is also an element of B. Notice that in particular, B is a subset of itself; a subset of B that isn't equal to B is called a proper subset. If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. A is called a subset if every element of A is an element of B.  B is a superset of A if every element in A is an element of B.

 

Unions, intersections, and relative complements

Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B. The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B. Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B. Symbolically, these are respectively

 

A ∪ B := {x : (x ∈ A) or (x ∈ B)};

A ∩ B := {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B} = {x ∈ B : x ∈ A};

A \ B := {x : (x ∈ A) and not (x ∈ B) } = {x ∈ A : not (x ∈ B)}.

 

A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all living dragons has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics.

 

Set Theory does have its place in a Business Organization.  Any organization essentially comprises of various types of resources such as men, machines, money, materials, etc.  The inter-relationship between these resources, often limited if not scarce, as also the inter-relationship between the subsets of each set of resources is used to equate assets of one kind with assets of another kind.  A subset of skilled workers within the set of all workers is a critical subset that has an impact on the overall productivity of the organization.  A subset of skilled salesmen within the set of all staff in the Marketing and Sales department of an organization is another critical subset, especially in a marketing driven organization which has an impact on the topline of the organization.  Similarly there could be subsets of products, materials, etc. which when inquired into analytically could pave the way for effective decision making and sound organizational plans, policies and procedures.