Sets

Introduction

Generally, a set is denoted by the capital letters of English alphabet A, B, C etc. and their members by small letters a,b,c etc. For example, the set of English vowels may be denoted by the capital letter V whereas its members a, e, i, o, u by small letters.
Expressing it in the form of the set
V = {a, e, i, o, u}

To indicate a member belonging to a set, symbol \(\in\) V, and is read as "a belongs to V"

To indicate that something doesn't belong to a set a symbol ∉ V, and is read as 'p doesn't belong to V'

The union of two sets is denoted by "U". For example, A U B is the union of two sets A and B. It is read as 'A union B'. This operation includes the two sets without repetition. The union of two sets A and B is A U B.

The intersection of two sets is denoted by ∩. For example, A ∩ B represents the intersection of A and B. It is read as 'A intersection B'. This operation includes the elements of the two sets belonging to both of them. If there is no any element common between two sets then A∩B = ø, where ø is an empty set. Here, the intersection set of two sets A and B is the newest set A ∩ B.

Description of Sets

1. Listing or rooster method: The elements are listed inside the brackets, { }. Eg: N = {1,2,3,4...}
2. Descriptive method: The common properties of elements of sets are described by words. Eg: N = {the counting numbers 1 and greater than 1}
3. Set builder or rule method: The elements are represented by a variable stating their common properties. Eg: N = {x: x ∈ N}

Types of Sets

Empty or Null Set: If a set contains no elements then, it is null set. It is denoted by { } or Φ. For example: A = {A set of cows with three legs} \(\therefore\)A = { }

Singleton or Unit Set: If a set contains only one elments then, it is called singleton set. For example: A = {The set of highest mountain in the world}

Finite Set: If a set contains finite number of elements i.e. countable collection then, it is called finite set. For example: A = {a set of even numbers less 20}

Infinite Set: Set containing uncountable or unlimited numbers/elments is known as infinite set. For example: A = {a set of all odd numbers}

Equal Set: Two sets having same elements are called equal sets. For example: A = {2, 4, 6, 8, 10} and B =n {the five multiples of 2} then, A = B

Equivalent Set: Two or more sets having same number of elements are called equivalent sets. It is denoted by '∼' sign. For example: If X = {1, 2, 3, 4, 5} and Y = {a, b, c, d, e}. Then n(X) = 5 and n(Y) = 5 \(\therefore\) X and Y are equivalent sets i.e.X ∼ Y.

Universal Set: A set which contains all the subsets of a set. For example: If E = { the set of even numbers} and O = {the set of odd numbers} , we can make these sets from the set W = {whole numbers} , we can make these sets from the set W = {whole numbers}. Therefore, W is the universal set for the two sets E and O.

Subset: The set made by the elements of the universal set is a subset of that universal set . For example a universal set U = {whole numbers from 1 to 30}
i.e U = {1, 2, 3.....29, 30}

Proper Subset: Let A and B be two sets where B is the subset of A. Then B is said to be the proper subset of A if B has at least one elements less than set A. it is denoted by B ⊂ A. For example: If A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4} then, B is said to be the proper subset of set A. \(\therefore\)B ⊂ A

Overlapping Set:Two sets having some elements in common are overlapping sets. For example : A ∩ B = {6} where 6 is common to both sets. Hence, A and B are called overlapping sets.

Disjoint set:If there is no element common between two sets then the sets are called disjoint sets. For example, the set of even numbers and the set off odd numbers up to 10 are disjoint sets.

Use of Venn-Diagram

In the 20th century, mathematician John Venn represented the operations on sets and subsets in a simple way by means of a figure. These figures for sets are given the name Venn diagrams after his name. Venn diagrams are basically used to solve verbal problems in mathematics.

Overlapping Set:

Two sets having some elements in common are overlapping sets. In the given Venn-diagram, 4, 5, 6 are common elements in both sets A and B, then A and B are said to be overlapping sets.

Disjoint Set:

If there is no element common between two sets then the sets are called disjoint sets. In the given Venn-diagram, there are no common elements for A and B.

Operations of Set:

There are mainly four operations of sets. They are:-
 1.) Union of set

Let U be the universal set and A and B be the subsets of U, The set of all members that belong to either setA or set B or both. A and B is the union of sets. It is denoted by A∪B.

In the given venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

\(\therefore\) A∪B = {1, 2 , 3 ,4, 5, 6, 7, 8, 9}

The shaded region on the Venn-diagram represents A∪B.

2.) Intersection of Set

Let U be the universal set and A and B be the subsets. The set of elements belonging to both sets A and B is the intersection of these set. It is denoted by A∩B.

In the given Venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

\(\therefore\) A∩B = {4, 5, 6}

The shaded region on the Venn-diagram represents A∩B.

3.) Complement of a set

Let U b the universal set and A be its subsets. Then the complement of set A, denoted by A^c or A' is the set of elements of U which do not belong to A.

In the given Venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

\(\therefore\) A' = {7, 8, 9, 10, 11, 12}

\(\therefore\) B' = {1, 2, 3, 10, 11, 12}

4.) Difference of Set

Let A and B be the subsets of the universal set U, then thedifferenceof two sets A and B is the set (a-) that represents the elements of A which are not in B.

The shaded region represents A-B.

\(\therefore\) A - B = A - (A∩B)

Also, the shaded region in the given Venn-diagram represents B - A.

\(\therefore\) B - A = B - (A∩B)

Cardinality of Sets

The number of elements in a set is known as a cardinal number. For example, if A = {a, b, c, d, e, f} then the cardinality of A is 6. It is written as n(A) = 6.

Here in the given Venn-diagram

U = {a, b, c, d, e, f, g, h, i, j}

n(U) = 10

A = {a, b, c, d, e, f}

n(A) = 6

B = {a, b, c, g, h}

n(B) = 5

If A and B are overlapping sets, the number of elements in non-overlapping parts A and B are denoted by n_o(A) and n_o(B) respectively.

n_o(A) represents the elements of A only.

n_o(B) represents the elements of B only.

n_o(A) = n(A - B) = n(A) - n(A∩B)

n_o(B) = n(B - A) = n(B) - n(A∩B)

In the above venn-diagram

n(A) = 6

n_o(A) = 3

n(B) = 5

n_o(B) = 2