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Set: Cardinality, Notations, Construction, Operations

In this tutorial, we will learn about the set cardinality, standard notations, construction of a set, and set operations with the help of examples. By Prerana Jain Last updated : May 09, 2023

Prerequisite: Set Theory

Cardinality of Set

It is the number of elements in a set denoted like, A= {1, 2, 3, 4}

|A| = 4  | | symbol of cardinality

Standard Notations of Set

There are many notations which are used in the set.

• C = set of all complex number that can be represented in form of a+ib, a, b are real number.
• R = set of all real number that is represented along a line.
• Q = set of all rational number that can be expression.
• Z = set of all integer.
• Number without fractional components.
• W = set of all whole number. Set of all positive (integer) and zero.
• N = Non negative number.

Construction of Set

A set can be represented by two methods:

1. Tabulation/Roster method
   In the tabulation method, we describe a set by actually writing every number of the set. In this method, we prepare a list of objects forming the set writing the elements one after another between a pair of curly brackets. Thus a set a whose elements are 1, 3, 5,... will be written as A = {1, 3, 5, ...}.
2. Set builder
   In this, we describe a set by actually writing the member of set the properties based on which reader can understand what is the set.
   A = { x: x E Z 0<x<5 }

Set Operations

In Discrete Mathematics, the set operations are performed on two or more sets. In a set theory, there are five major types of operations performed on sets, such as: Union of sets (∪), Intersection of sets (∩), Difference of sets ( – ), Complement of sets ('), and Symmetric difference of sets.

1. Union of sets

For any two set A and B, the union of A and B written as AUB is the set of all elements which are members of the set A or the set B or both, Symbolically it is written as: A U B = { x:x E A or X E B }

Example:

Let,
A= {1, 2, 3, 4}
B = { 2, 4, 6, 8, 10}

Then, 
A U B = {1, 2, 3, 4, 5, 6, 7, 8, 10}

2. Intersection of sets

The intersection of two set A and B denoted A intersection B is the set of elements which belongs to both A and B, it is written as: A intersection B = { x:x E A and xEB}

3. Difference of sets

The difference of two sets A and B in that order is the set of elements which belongs to A, but which O does not belong to B. We denote the difference of A and B by: A – B or A ~ B

Which reads as "A difference B" or "A minus B" symbolically A - B = {x:x E A and X does not belong to B}

A ~ B is also called the compliment of B with respect to A.

Example:

Let,
A = {a, b, c, d, e}
B = {f, b, d, g}
A – B = {a, c, e}
B – a = {f.g}
A – B does not equal to B – A.

4. Complement of sets

If U is a universal set containing A then U-A is called the complement of A and is denoted by A' or Ac thus
A' = U – A = {x:x E U and x does not belongs to U}
A' = { x:x does not belong to A}

Example:

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

Then,
A' = {1, 3, 5, 7}

5. Symmetric difference of sets

If A and B are two sets we define their symmetric difference as the set of all elements that belong to A or to B but not to both A and B and we denote it by (A - B)U(B – A).

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