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Subject : Mathematics
Standard : 9 nth std
Unit : Sets .

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Introduction of sets, Definition, Examples , Representation of Set, Types of set, Operation on sets , Properties of sets.

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The concept of sets is used for the foundation of various topics in mathematics.

To learn sets we often talk about the collection of objects, such as a set of vowels, set of negative numbers, a group of friends, a list of fruits, a bunch of keys, etc.

Definition
sets are a collection of well-defined objects or elements.
A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}.
For example, 1)set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}.
2). A set of vowels,  A = { a, e , i, o, u}

3) { 2,4,6,8,10,12}.    etc....

https://images.app.goo.gl/wdJEZ3jfkbdLanGN8

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1) Roaster method
2) Set builder method

1) Listing/ Roaster Method:  In this method all or partial members of the set are listed.

Examples:

a)  Let R be the set of Natural number less than 10.

{1,  2,  3,  4,  5,   6,  7,  8,9}

2) . Set builder method : In this method the set is described by listing the properties that describe the  elements of the set.

Example:  1) P is a set of counting numbers greater than 12
P = {x : x is a counting number and greater than 12}

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1) Empty set
2) Singleton Set or Unit set
3) Finite Set
4) Infinite Aste
5) Universal Set
6) Sub set

Finite Set

A set which contains a definite number of elements is called a finite set.
Example − S = { x | x ∈ N and 70 > x > 50 }

Infinite Set

A set which contains infinite number of elements is called an infinite set.
Example − S = { x | x ∈ N and x > 10 }

Subset

A set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y.
Example 1 − Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y ⊆ X.

Universal Set

It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.

Empty Set or Null Set

An empty set contains no elements. It is denoted by ∅.
Example − S = { x | x ∈ N and 7 < x < 8 } = ∅

Singleton Set or Unit Set

Singleton set or unit set contains only one element. A singleton set is denoted by { s }.

Example − S = { x | x ∈ N, 7 < x < 9 } = { 8 }

Equal Set

If two sets contain the same elements they are said to be equal.
Example − If A = { 1, 2, 6 } and B = { 6, 1, 2 }, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

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Types of sets | Finite set | Infinite set | Empty set | Singleton set| subset | Math Dot Com

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Unions
An element is in the union of two sets if it is in the first set, the second set, or both. The symbol we use for the union is
∪∪. The word that you will often see that indicates a union is "or".

Example
1. Union of Two setsA={2,5,7,8} and B={1,4,5,7,9}

Find  A∪B
We include in the union every number that is in A or is in B:
A∪B={1,2,4,5,7,8,9}

Intersections
An element is in the intersection of two sets if it is in the first set and it is in the second set. The symbol we use for the intersection is
∩∩. The word that you will often see that indicates an intersection is "and".
Example  : Intersection of Two sets
A={3,4,5,8,9,10,11,12} and B= {5,6, 7,8,9} Find
A∩BA∩B.
Solution
We only include in the intersection that numbers that are in both A and B:
A∩B={5,8,9}

https://courses.lumenlearning.com/math4libarts/chapter/union-intersection-and-complement/

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Union and Intersection of Sets | Venn Diagrams | Set Symbols | Intervals [Animated] - Pre-Algebra

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Property 1. Commutative property

Intersection and union of sets satisfy the commutative property.

A⋂B = B⋂A

A⋃B = B⋃A

https://youtu.be/fxWbmGNJ1Z4

Property 2. Associative property

Intersection and union of sets satisfy the associative property.

(A⋂B)⋂C = A⋂(B⋂C)

(A⋃B)⋃C = A⋃(B⋃C)

https://youtu.be/fxWbmGNJ1Z4

https://youtu.be/ly-T6TyEyDk

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1) If P = {1,2,5,7,9}, Q = {2, 3,5,9,11}, R = {3, 4,5,7,9} and S = {2, 3, 4,5, 8}, then find (i) (P U Q) U R (ii) (P n Q) n S (iii) (Q n S) n R

Solution :

P = {1, 2, 5, 7, 9}
Q = {2, 3, 5, 9, 11}
R = {3, 4, 5, 7, 9}
S = {2, 3, 4, 5, 8}

(i) (P U Q) U R

(P U Q)  =  {1, 2, 3, 5, 7, 9, 11}
(P U Q) U R  =  {1, 2, 3, 4, 5, 7, 9, 11}  -------(1)

(ii) (P n Q) n S

P n Q  =  {2, 5, 9}

(P n Q) n S =  {2, 5}  ------(2)

(iii) (Q n S) n R
Q n S  =  {2, 3, 5}
(Q n S) n R =  {3, 5}  ------(3)

https://www.math-only-math.com/worksheet-on-properties-of-a-set.html

https://www.onlinemath4all.com/practice-
problems-on-properties-of-set-theory.html

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MATHEMATICS | 1st PUC | CH 01 | SETS - PROBLEMS ON PROPERTIES OF SETS | S12

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Set Operations

The set operations are performed on two or more sets to obtain a combination of elements, as per the operation performed on them. In a set theory, there are three major types of operations performed on sets, such as:

1. Union of sets (∪)

2. Intersection of sets (∩)

3. Difference of sets ( – )

Let us discuss these operations one by one.

Union of Sets

If two sets A and B are given, then the union of A and B is equal to the set that contains all the elements, present in set A and set B. This operation can be represented as;

A ∪ B = {x: x ∈ A or x ∈ B}

Where x is the elements present in both the sets A and B.

Example: If set A = {1,2,3,4} and B {6,7}

Then, Union of sets, A ∪ B = {1,2,3,4,6,7}

Intersection of Sets

If two sets A and B are given, then the intersection of A and B is the subset of universal set U, which consist of elements common to both A and B. It is denoted by the symbol ‘∩’. This operation is represented by:

A∩B = {x : x ∈ A and x ∈ B}

Where x is the common element of both sets A and B.

The intersection of sets A and B, can also be interpreted as:

A∩B = n(A) + n(B) – n(A∪B)

Where,

n(A) = cardinal number of set A,

n(B) = cardinal number of set B,

n(A∪B) = cardinal number of union of set A and B.

Example: Let A = {1,2,3} and B = {3,4,5}

Then, A∩B = {3}; because 3 is common to both the sets

Difference of Sets

If there are two sets A and B, then the difference of two sets A and B is equal to the set which consists of elements present in A but not in B. It is represented by A-B.

Example: If A = {1,2,3,4,5,6,7} and B = {6,7} are two sets.

Then, the difference of set A and set B is given by;

A – B = {1,2,3,4,5}