Set Theory

What is Set?

A set is a collection of well defined and different objects.

Some standard sets

• N : Set of all natural numbers.
• W : Set of all whole numbers.
• I or Z : Set of all integers.
• Q : Set of all rational numbers.
• R : Set of all real numbers.

Methods of Designating a set

1. Tabular, Roaster or Enumeration Method : represent a set by listing all elements. For e.g. A = {a,e,i,o,u}.
2. Selector, Set-builder or Rule Method : represented by specifyiung the defining property. For e.g. A = {x : x is a vowel in English alphabets}.

Types of Sets

1. Finite Set : finite number of elements.
2. Infinite Set : infinite number of elements.
3. Singleton/Unit Set : set with only one element.
4. Empty/Null/Void Set : set with no elements and is denoted by {} or φ .
5. Subset : If every element of A is in B then A is subset of B and it is denoted as A ⊂ B.
6. Superset : If every element of A is in B then B is superset of A and it is denoted as B ⊃ A.
7. Proper Set : A ⊂ B and A ≠ B.

   φ and A are called improper subsets of A.

8. Power Set : set of all subsets.
9. Universal Set : set of all elements.

   or

   if all the sets are subsets of U then U is a universal set.

Important Terms

• Comparable Set : if one of the two sets is a subset of the other then they are comparable sets.
• Equality of Sets : A ⊂ B and B ⊂ A or if boths sets have the same elements. It is denoted as A = B.
• Non Comparable Set : if no set of the two sets is a subset of the other then they are non comparable sets.
• Order of a Finite Set : number of different elements of a finite set and is denoted by O(set_name). For e.g O(A) = 4;.
• Cardinality : Number of different elements in a set.
• Equivalent Sets : if O(A) = O(B) then both are equivalent.

Venn Diagrams

A Venn diagram is a diagrammatic representation of sets where each set is represented by a circle.

Operations on Sets

1. Union of Sets : The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. It is denoted by A ∪ B.
   
2. Intersection of Sets : The intersection of two sets A and B is the set of elements which are in both A and B. It is denoted by A ∩ B.
   
3. Difference of Sets : The difference of the set B from the set A is the set of all elements that are in A but not in B. It is denoted by A - B or A\B.
   
4. Complement of a Set : The complement of a set A refers to elements not in A. It is denoted by A' or A̅.
   

Fundamental Laws of Set Theory

Inclusion - Exclusion Principle

Number of elements of a finite set A is denoted by n(A).

Following results of number of elements should be kept in mind for doing problems :

1. n(A ⋃ B) = n(A) + n(B) - n(A ⋂ B)
2. n(A ⋃ B) = n(A) + n(B) if A, B are disjoint sets.
3. n(A ⋃ B) = n(A-B) + n(B-A) - n(A ⋂ B)
4. n(A) = n(A-B) + n(A ⋂ B)
5. n(B) = n(B-A) + n(A ⋂ B)
6. n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) - n(A ⋂ B) - n(A ⋂ C) - n(B ⋂ C) + n(A ⋂ B ⋂ C)
7. n(A' ⋃ B') = n((A ⋂ B)') = n(U) - n(A ⋂ B)
8. n(A' ⋂ B') = n((A ⋃ B)') = n(U) - n(A ⋃ B)
9. n(A ⋂ B' ⋂ C') = n(A) - n(A ⋂ B) - n(A ⋂ C) + n(A ⋂ B ⋂ C)

Cartesian Product of Sets

The Cartesian product of two sets A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

For example, if A = {1, 2} and B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}.

Partitions of Sets

Minimum Set/Minset/Minterm

A minset of a set A with respect to a set function f is a subset B of A such that for every subset C of A, if f(B) = f(C), then B is a subset of C.

Maximum Set/Maxset/Maxterm

A maxset of a set A with respect to a set function f is a subset B of A such that for every subset C of A, if f(B) = f(C), then C is a subset of B.