Days of week, Months of a year, All prime number less than 10, All even numbers less than 15 these are well defined it will not change from one person to another.So, these are the example of Sets.

Best actor or actress in Bollywood, Best cricketers of India, Most dangerous animal of the world, Most poisonous animals in the world these are not well defined as it will varies from one person to another.

If you ask some of your friends to make a list of Best actor or actress in Bollywood there is a chance of variation of list as the criteria of choosing best actor or actress are not same for all persons. Similarly it will happen in other cases.

So, these are not sets.

How a set is represented?

There are 2 methods to represent a set

1. Roster or Tabular Form: In this form, we list all the elements of the set within braces { } and separate by commas.

A set of natural numbers less than 10 A={1,2,3,4,5,6,7,8,9}

2. Set-builder Form: In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

A={ n:n is an natural number less than 10} or A={ n:n is a natural number <10}

Elements - The objects in a set are called the elements (or members ) of the set; the elements are said to belong to the set (or to be in the set), and the set is said to contain the elements. Usually the elements of a set are other mathematical objects, such as numbers, variables, or geometric points. ### Types of Sets

1. Empty Sets: A set without any element is known as an empty set or the void set or null set. It is denoted by Φ or {}.

2. Singleton Set: A set with a single element, is known as a singleton set.

3. Finite and Infinite Set: A set which has a finite number of elements, is called a finite set, if not the set is known as an infinite set.

4. Equal Sets: Two sets A & B are said to be equal, if all elements of A is also an element of B, i.e. two equal sets will have the same elements.

5. Equivalent Sets: Two finite sets A & B are said to be equal if number of elements in both sets are equal, i.e. n(A) = n(B)

6. Subsets: Set A will be a subset of Set B if every element of Set A is also present in Set B. In simple words, set A is contained inside Set B.

For Example: If set A has {A, B} and set B has {A, B, C}, then A will be the subset of B since elements of A are also present in set B.

7. Superset: If A is a subset of B (A ⊂ B) then B is the superset of A (B ⊃ A).

8. Power Set: The collection of all of the subsets of a set A is known as the power set of A. Generally this is denoted by P(A). If the no of elements in A i.e. n(A) = n, then the no of elements in P(A) = 2n.

9. Universal Set: A set which contains all sets in a given context is defined as the universal set.

10. Proper Subset - Set A is considered to be a proper subset of Set B if Set B contains at least one element that is not present in Set A.

If A={1, 2, 3}, B={1, 3, 2, 4}then set A is the proper subset of B because 4 is not present in the set A.

11. Improper Set- A subset which contains all the elements of the original set is called an improper subset.

Set A ={2,5,7} Then, the subsets of P are;

{}, {2}, {5}, {7}, {2,5}, {5,7}, {2,7} and {2,5,7}.

Where, {}, {2}, {5}, {7}, {2,5}, {5,7}, {2,7} are the proper subsets and {2,4,6} is the improper subsets. Therefore, we can write {2,5,7} ⊆ A.

12. Disjoint Sets: If two sets A and B have no any common elements i.e. A∩ B=Φ then they are called disjoint sets.

13. Venn-Diagrams: Venn diagrams are diagrams that show the relationship between two sets. The universal set U is represented by a rectangle, and its subsets are represented by closed curves (circle) within the rectangle in Venn diagrams.

Venn Diagram

14. Union of Sets: The union of two sets A & B is denoted by A ∪ B will be the set of all those elements which are either in set A or in set B or in both A and B.

15. Intersection of Sets: The intersection of two sets A & B is denoted by A ∩ B, and it is the set of all elements which are common in both set A and set B.

16. Difference of Sets: The difference between set A and set B is such that it has only those elements which are in the set A and not in the set B. A– B = {p : p ∊ A and p ∉ B}

Similarly, B – A = {p: p ∊ B and p ∉ A}.

17. Complement of a Set: Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U - A), is defined as

A’ = {x U : x A}

X A’ x A### Some important symbols used in this chapter

∪ - Union∩ - IntersectionA' or - Complement (Element not in A)⊆ - Subset or Improper subset⊂ - Proper Subset⊄ - Not a Subset⊇ - Superset⊃ - Proper Superset⊅ - Not a Superset∈ - Element of∉ - Not element ofØ - Empty set or Null set| - Such that: - Such that∀ - For All∃ - There Exists∴ - ThereforeN - Natural NumbersZ - IntegersQ - Rational NumbersA - Algebraic NumbersR - Real NumbersI - Imaginary NumbersC - Complex Numbers### Laws

Idempotent Laws: For any set A

A ∪ A = AA ∩ A = A

Identity Laws: For any set A

A ∪ Φ = AA ∩ U = A

Commutative Laws: For any two sets A and B

A ∪ B = B ∪ AA ∩ B = B ∩ A

Associative Laws: For any three sets A, B and C

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

Distributive Laws: If A, B and C are three sets

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

De-Morgan’s Laws: If A and B are two sets

(A ∪ B)’ = A’ ∩ B’(A ∩ B)’ = A’ ∪ B’### Some formulae

n(A ∪ B) = n(A) + n (B) – n(A ∩ B)If (A ∩ B) = Φ, then n (A ∪ B) = n(A) + n(B)n(A – B) = n(A) – n(A ∩ B)n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) Try to solve Practice Paper

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1. Roster or Tabular Form: In this form, we list all the elements of the set within braces { } and separate by commas.

A set of natural numbers less than 10 A={1,2,3,4,5,6,7,8,9}

2. Set-builder Form: In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

A={ n:n is an natural number less than 10} or A={ n:n is a natural number <10}

Elements - The objects in a set are called the elements (or members ) of the set; the elements are said to belong to the set (or to be in the set), and the set is said to contain the elements. Usually the elements of a set are other mathematical objects, such as numbers, variables, or geometric points.

### Types of Sets

1. Empty Sets: A set without any element is known as an empty set or the void set or null set. It is denoted by Φ or {}.

2. Singleton Set: A set with a single element, is known as a singleton set.

3. Finite and Infinite Set: A set which has a finite number of elements, is called a finite set, if not the set is known as an infinite set.

4. Equal Sets: Two sets A & B are said to be equal, if all elements of A is also an element of B, i.e. two equal sets will have the same elements.

5. Equivalent Sets: Two finite sets A & B are said to be equal if number of elements in both sets are equal, i.e. n(A) = n(B)

6. Subsets: Set A will be a subset of Set B if every element of Set A is also present in Set B. In simple words, set A is contained inside Set B.

For Example: If set A has {A, B} and set B has {A, B, C}, then A will be the subset of B since elements of A are also present in set B.

7. Superset: If A is a subset of B (A ⊂ B) then B is the superset of A (B ⊃ A).

8. Power Set: The collection of all of the subsets of a set A is known as the power set of A. Generally this is denoted by P(A). If the no of elements in A i.e. n(A) = n, then the no of elements in P(A) = 2n.

9. Universal Set: A set which contains all sets in a given context is defined as the universal set.

10. Proper Subset - Set A is considered to be a proper subset of Set B if Set B contains at least one element that is not present in Set A.

If A={1, 2, 3}, B={1, 3, 2, 4}

then set A is the proper subset of B because 4 is not present in the set A.

11. Improper Set- A subset which contains all the elements of the original set is called an improper subset.

Set A ={2,5,7} Then, the subsets of P are;

{}, {2}, {5}, {7}, {2,5}, {5,7}, {2,7} and {2,5,7}.

Where, {}, {2}, {5}, {7}, {2,5}, {5,7}, {2,7} are the proper subsets and {2,4,6} is the improper subsets. Therefore, we can write {2,5,7} ⊆ A.

12. Disjoint Sets: If two sets A and B have no any common elements i.e. A∩ B=Φ then they are called disjoint sets.

13. Venn-Diagrams: Venn diagrams are diagrams that show the relationship between two sets. The universal set U is represented by a rectangle, and its subsets are represented by closed curves (circle) within the rectangle in Venn diagrams.

Venn Diagram

14. Union of Sets: The union of two sets A & B is denoted by A ∪ B will be the set of all those elements which are either in set A or in set B or in both A and B.

15. Intersection of Sets: The intersection of two sets A & B is denoted by A ∩ B, and it is the set of all elements which are common in both set A and set B.

16. Difference of Sets: The difference between set A and set B is such that it has only those elements which are in the set A and not in the set B. A– B = {p : p ∊ A and p ∉ B}

Similarly, B – A = {p: p ∊ B and p ∉ A}.

17. Complement of a Set: Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U - A), is defined as

A’ = {x U : x A}

X A’ x A

### Some important symbols used in this chapter

∪ - Union

∩ - Intersection

A' or - Complement (Element not in A)

⊆ - Subset or Improper subset

⊂ - Proper Subset

⊄ - Not a Subset

⊇ - Superset

⊃ - Proper Superset

⊅ - Not a Superset

∈ - Element of

∉ - Not element of

Ø - Empty set or Null set

| - Such that

: - Such that

∀ - For All

∃ - There Exists

∴ - Therefore

N - Natural Numbers

Z - Integers

Q - Rational Numbers

A - Algebraic Numbers

R - Real Numbers

I - Imaginary Numbers

C - Complex Numbers

### Laws

Idempotent Laws: For any set A

A ∪ A = A

A ∩ A = A

Identity Laws: For any set A

A ∪ Φ = A

A ∩ U = A

Commutative Laws: For any two sets A and B

A ∪ B = B ∪ A

A ∩ B = B ∩ A

Associative Laws: For any three sets A, B and C

A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive Laws: If A, B and C are three sets

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De-Morgan’s Laws: If A and B are two sets

(A ∪ B)’ = A’ ∩ B’

(A ∩ B)’ = A’ ∪ B’

### Some formulae

n(A ∪ B) = n(A) + n (B) – n(A ∩ B)

If (A ∩ B) = Φ, then n (A ∪ B) = n(A) + n(B)

n(A – B) = n(A) – n(A ∩ B)

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)

Try to solve Practice Paper

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