Sets

1.0Definition of Sets 

In Mathematics, a set is a well-defined collection of different objects, considered as an object in its own right. These objects can be anything: numbers, people, letters, or other abstract entities. Sets are denoted by curly braces, such as {1, 2, 3}, and each object in a set is called an element or member. Sets are fundamental in Mathematics and are used to study relationships, perform operations, and define various Mathematical structures.

2.0Representation of Sets in Set Theory

In set theory, sets are typically represented using curly braces { } to enclose their elements. The elements of a set can be anything, such as numbers, letters, or even other sets. Here are some common ways sets are represented and described in set theory:

1. Roster Form

In roster form, the elements of a set are listed explicitly within curly braces, separated by commas. For example: Set P = {1, 2, 3, 4, 5}

2. Set-Builder Form

In set-builder notation, a set is described by specifying a property or condition that its elements must satisfy. The notation is {x | P(x)}, where x represents the elements and P(x) is the property or condition. For example: Set B = {x | x is an even number}

3. Interval Notation 

In Mathematics, intervals can be used to represent sets of real numbers. For example, Set C = {x | 0 < x < 10} can be represented as (0, 10), indicating all real numbers greater than 0 and less than 10.

4. Special Sets

• Empty Set: Denoted by ϕ or { }, it is the set with no elements.
• Universal Set: Denoted by U, it is the set containing all elements under consideration.
• Singleton Set: A set containing only one element. For example, {7}.

These representations help in defining and describing sets in a clear and concise manner, allowing for precise Mathematical reasoning and operations within set theory.

3.0Elements of Set

Elements of a set are the individual objects or numbers that belong to the set. In set notation, elements are listed inside curly braces { } and separated by commas. The elements of a set can be letters, numbers, symbols, or even other sets. Each element in a set is unique, and there are no duplicate elements within a set.

For example, consider the set A = {1, 2, 3, 4}. In this set:

• 1, 2, 3, and 4 are the elements of set A.
• Each element is distinct and appears only once in the set.
• The order of elements in a set does not matter. Thus, {1, 2, 3, 4} is the same as {4, 3, 2, 1} as they contain the same elements.

In set theory, elements are crucial for defining the properties and relationships of sets, performing set operations, and analyzing mathematical concepts.

4.0Types of Sets

1. Null Set or Empty Set or Void Set

A set having no elements is called a Null Set or Empty Set or Void set. It is denoted by ϕ or { }.

2. Singleton Set

A set consisting of one or a single element is called a Singleton Set.

3. Finite Set

A set that has only a finite number of elements is called a Finite Set.

4. Infinite Set

A set which has an infinite number of elements is called an infinite Set.

5. Subset

If every element in set A is also found in set B, then A is termed as a subset of B, denoted as A ⊆ B.

6. Proper subset

If set A is a subset of set B, but A is not equal to B, then A is considered a proper subset of B, represented as A ⊂ B.

7. Universal Set

A set consisting of all possible elements that occur in the discussion is called a Universal set and is denoted by U.

8. Power set

Let A be any set. The set of all subsets of A is called the power set of A and is denoted by P(A).

9. Equivalent Set or Equal Set

If every element of set A is also an element of set B, and vice versa, they are equal sets denoted as A = B. 

For example, A = {3,4,5,6} and B = {6,5,4,3} are equal sets. However, A = {set of even numbers} and B = {set of natural numbers} are not equal as their elements differ.

10. Disjoint Sets

Two sets, X and Y are called disjoint sets if they have no common elements, denoted as X ∩ Y = ϕ. This signifies that their intersection results in zero elements.

5.0Operations on Sets 

1. Cardinality of a Set

The cardinality of a set A, denoted as |A|, represents the number of elements in the set.

Formula: ∣A∣ = n, where n is the count of elements in set A.

2. Union of Sets

The union of two sets, denoted as A ∪ B, comprises all the elements that are in either set A or set B, or both.

Formula : A ∪ B = { x : x ∈ A or x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A ∪ B = {1,2,3,4,5}.

3. Intersection of Sets

The intersection of two sets, A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.

Formula : A ∩ B = { x : x ∈ A and x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A∩B = {3}.

4. Complement of Sets

The complement of a set A, denoted as A′ or Ā, with respect to a universal set U, contains all elements in U that are not in A.

Formula : A′ = { x ∈ U : x ∉ A}

Example: If U = {1,2,3,4,5} is the universal set and A = {1,2}, then A′ = {3,4,5}.

5. Difference of Sets

The set difference between sets A and B, denoted as A−B, is the set of elements that are in A but not in B.

Formula : A - B = { x : x ∈ A and x ∉ B}

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}.

6. Cartesian Product of Sets

The Cartesian product of Set A and Set B, denoted as A × B, represents the collection of all ordered pairs (a, b) where a is in Set A and b is in Set B.

 Formula : A × B = { (a, b) : a ∈ A, b ∈ B}

7. Power Set

The power set of a set A, denoted as P(A), is the set of all subsets of A, including the empty set and A itself.

Formula : P(A) = { B : B ⊆ A}

These operations are fundamental in set theory and are used to manipulate sets, study relationships between sets, and solve various Mathematical problems.

6.0Set Formulas 

Some of the Important Sets Formula are

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint sets.
•  n( A – B) + n( A ∩ B ) = n(A)
• n( B – A) + n( A ∩ B ) = n(B)
• n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )
• n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

7.0Venn Diagrams

Clearly (A – B) ∪ (B – A) ∪ (A ∩ B) = A ∪ B

8.0Properties of Sets 

1. Associative Property

• A ∪ ( B ∪ C) = ( A ∪ B) ∪ C
• A ∩ ( B ∩ C) = ( A ∩ B) ∩ C

2. Distributive Property

• A ∪ ( B ∩ C) = ( A ∪ B) ∩ ( A ∪ C)
• A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C)

3. De Morgan’s Law

• Law of union : ( A ∪ B)′ = A′ ∩ B′ 
• Law of Intersection :( A ∩ B)′ = A′ ∪ B′

4. Complement Law

• A ∪ A′ = A′ ∪ A = U
• A ∩ A′ = ϕ

5. Idempotent Law and Law of a null and universal set

For any Finite set, A

• A ∪ A = A
• A ∩ A = A
• Φ′ = U
• ϕ = U′

9.0Solved Examples

Example 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between –3 and 7

Solution: Roaster Form: { –2, –1, 0, 1, 2, 3, 4, 5, 6}

     Set Builder Form: {x : x ∈ Z, –3 <  x < 7} 

Example 2:  Given the sets X = {1, 2, 3}, Y = {3, 4}, Z = {4, 5, 6}, then X ∩ ( Y ∪ Z) is

Solution: Y ∪ Z = {3, 4, 5, 6}

X ∩ ( Y ∪ Z) = {3}

Example 3: A group of members know at least one of the two languages, Hindi or Kannada. In the group, 150 members know Hindi, 80 members know Kannada and 70 of them know both. How many members are there in the group?

Solutions: Let H = Set of persons who know Hindi.

K = Set of persons who know Kannada.

n(H ∩ K) = the number of persons who know both Hindi and Kannada is 70.

n(H ∪ K) = n(H) + n(K) – n(H ∩ K)

= 150 + 80 – 70 = 160

Example 4: Find A ✕ B when A = {x | x is prime number less than 5} and B = {1, 2, 3}

Solution: A = {2, 3}; B = {1, 2, 3}

      A ✕ B = {(2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

10.0Practice Problems

Question 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between 5 and 15.

Question 2:  Given the sets A = {5, 2, 7}, B = {3, 8}, C = {3, 2, 6}, then find A ∩ ( B ∪ C).

Question 3: In a class of 60, students like Mathematics, 25 like science, and 15 like both. Then the number of students who like either Mathematics or Science is?

Question 4: Let A and B be the two sets in the universal set. Then A – B equals-

A ∩ B′ b. A′ ∩ B c. A ∩ B d. None of these