__ What is a Set?
__A set is a well-defined collection of distinct objects.

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

__What is an element of a Set?__

- The objects in a set are called its elements.
- So in case of the above Set A, the elements would be 1, 2, 3, 4, and 5. We can say, 1 ∈ A, 2 ∈ A

- Usually we denote Sets by CAPITAL LETTERs like A, B, C, etc. while their elements are denoted in small letters like
*x, y, z* - If
*x*is an element of A, then we say*x*belongs to A and we represent it as*x*∈ A - If
*x*is not an element of A, then we say that*x*does not belong to A and we represent it as*x*∉ A

__How to describe a Set?__

- Roaster Method or Tabular Form
- In this form, we just list the elements
- Example A = {1, 2, 3, 4} or B = {
*a, b, c, d, e*}

- Set- Builder Form or Rule Method or Description Method
- In this method, we list the properties satisfied by all elements of the set

Example *A = {x : x ∈ N, x< 5}*

Some examples of Roster Form vs Set-builder Form

__Sets of Numbers__

- Natural Numbers (N)

N = {1, 2, 3,4 ,5 6, 7, …}

- Integers (Z)

Z = {…, -3, -2, -1, 0, 1, 2, 3, 4, …}

- Whole Numbers ( W )

W = {0, 1, 2, 3 4, 5, 6…}

- Rational Numbers (Q)

*{p/q : p ∈ Z, q ∈ Z, q ≠0}*

__Finite Sets & Infinite Sets__

Finite Set: A set where the process of counting the elements of the set would surely come to an end is called finite set

- Example: All natural numbers less than 50
- All factors of the number 36

Infinite Set: A set that consists of uncountable number of distinct elements is called infinite set.

Example: Set containing all natural numbers *{x | x ∈ N, x > 100} *

__Cardinal number of Finite Set__

The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by *n*(A)

- Example
*A = {1, 2, 3, 4}*then*n(A) = 4* *A = {x | x is a letter in the word ‘APPLE’}.*Therefore*A = {A, P, L, E}*and*n(A) = 4**A = {x | x is the factor of 36},*Therefore*A = { 1, 2, 3, 4, 6, 9, 12, 18, 36}*and*n(A) = 9*

__Empty Set__

- A set containing no elements at all is called an empty set or a null set or a void set.
- It is denoted by ϕ (phai)
- In roster form you write ϕ = { }
- Also n (ϕ) = 0
- Examples:
*{x | x ∈ N, 3 < x < 4} = ϕ* - {
*x*|*x*is an even prime number,*x*> 5} = ϕ

- Examples:

__Non Empty Set__

- A set which has at least one element is called a non-empty set
- Example: A = {1, 2, 3} or B = {1}

__Singleton Set__

- A set containing exactly one element is called a singleton set
- Example: A = {
*a*} or B = {1}

- Example: A = {

__Equal Sets__

- Two set A and B are said to be equal sets and written as A = B if every element of A is in B and every element of B is in A
- Example A = {1, 2, 3, 4} and B = {4, 2, 3, 1}

- It is not about the number of elements. It is the elements themselves.
- If the sets are not equal, then we write as A ≠ B

__Equivalent Sets__

- Two finite sets A and B are said to be equivalent, written as A ↔ B, if n(A) = n(B), that is they have the same number of elements.
- Example: A = {a, e, i, o, u} and B = {1, 2, 3, 4, 5}, Therefore n(A) = 5 and n(B) = 5 therefore A ↔ B

- Note: Two equal sets are always equivalent but two equivalent sets need not be equal.

__Subsets__

- If A and B are two sets given in such a way that every element of A is in B, then we say A is a subset of B and we write it as A ⊆ B
- Therefore is A ⊆ B and
*x ∈ A*then*x ∈ B* - If A is a subset of B, we say B is a super set of A and is written as B ⊇ A
- Every set is a subset of itself.
- i.e. A ⊆ A, B ⊆ B etc.

- Empty set is a subset of every set
- i.e. ϕ ⊆ A, ϕ ⊆ B

- If A ⊆ B and B ⊆ A, then A = B
- Similarly, if A = B, then A ⊆ B and B ⊆ A
- If set A contains
*n*elements, then there are 2subsets of A^{n}

__Power Set__

- The set of all possible subsets of a set A is called the power set of A, denoted by P(A). If A contains
*n*elements, then P(A) = 2^{n}- i.e. if A = {1, 2}, then P(A) = 2
^{2}= 4 - Empty set is a subset of every set
- So in this case the subsets are {1}, {2}, {2, 3} & ϕ

- i.e. if A = {1, 2}, then P(A) = 2

__Proper Subset__

Let A be any set and let B be any non-empty subset. Then A is called a proper subset of B, and is written as A ⊂ B , if and only if every element of A is in B, and there exists at least one element in B which is not there in A.

- i.e. if A ⊆ B and A ≠ B then A⊂ B
- Please note that ϕ has no proper subset
- A set containing n elements has (2n – 1) proper subsets.
- i.e. if A = {1, 2, 3, 4}, then the number of proper subsets is (2
^{4}– 1) = 15

- i.e. if A = {1, 2, 3, 4}, then the number of proper subsets is (2

__Universal Set__

- If there are some sets in consideration, then there happens to be a set which is a super set of each one of the given sets. Such a set is known as universal set, to be denoted by U or ξ .
- i.e. if A = {1, 2}, B = {3, 4}, and C = {1, 5}, then U or ξ = {1, 2, 3, 4, 5}

__Operations on Sets__

__Union of Sets__

- The union of sets A and B, denoted by A∪ B , is the set of all those elements, each one of which is either in A or in B or in both A and B
- If there is a set A = {2, 3} and B = {
*a, b*}, then A∪ B = {2, 3,*a, b*} - So if
*A∪ B = {x | x ∈ A or x ∈ B}*then x ∈ A ∪ B which means*x ∈ A or x ∈ B* - And if
*x*∉ A ∪ B which means*x ∉ A or x ∉ B*

__Interaction of Sets__

- The intersection of sets A and B is denoted by A ∩ B , and is a set of all elements that are common in sets A and B.
- e. if A = {1, 2, 3} and B = {2, 4, 5}, then A ∩ B = {2} as 2 is the only common element.
- Thus A ∩ B = {
*x : x*∈ A and*x*∈ B} then x ∈ A ∩ B i.e. x ∈ A and x ∈ B - And if x ∉ A ∩ B i.e. x ∉ A and x ∉ B

Disjointed Sets

- Two sets A and B are called disjointed, if they have no element in common. Therefore:
- x ∉ A ∩ B i.e. x ∉ A and x ∉ B

Intersecting sets

- Two sets are said to be intersecting or overlapping or joint sets, if they have at least one element in common.
- Therefore two sets A and B are overlapping if and only if A ∩ B ≠ ϕ
- Intersection of sets is Commutative
- i.e. A ∩ B = B ∩ A for any sets A and B

- Intersection of sets is Associative i.e. for any sets, A, B, C,
- (A ∩ B) ∩ C = A ∩ ( B ∩ C)
- A ⊆ B, then A ∩ B = A
- Since A ⊆ ξ, so A ∩ ξ = A

- For any sets A and B we have
- A ∩ B ⊆ A and A ∩ B ⊆ B
- A ∩ ϕ = ϕ for every set A

__Difference of Sets__

- For any two sets A and B, the difference A – B is a set of all those elements of A which are not in B.

i.e. if A = {1, 2, 3, 4, 5} and B = {4, 5, 6}, Then A – B = {1, 2, 3} and B – A = {6}

Therefore A – B = { x | x ∈ A and x ∉ B}, then x ∈ A – B then x ∈ A but x ∉ B

- If A ⊆ B then A – B = ϕ

__Complement of a Set__

Let x be the universal set and let A ⊆ x. The the complement of A, denoted by A’ is the set if all those elements of x which are not in A.

- i.e. let ξ = {1, 2, 3, 4, 5, 6, 7, 8} and A= {2, 3, 4}, then A’={1, 5, 6, 7, 8}
- Thus A’={x | x ∈ ξ and x ∉ A} clearly x ∈A’ and x ∉ A

Please note

- ϕ’ = ξ and ξ’ = ϕ
- A ∪ A’ = ξ and A ∩ A’ = ϕ

__Disruptive laws for Union and Intersection of Sets__

For any three sets , B, C, we have the following

- A ∪ ( B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Say A = {1,2}, B = {2, 3} and C = {3, 4}

Therefore A ∪ ( B ∩ C) = {1, 2, 3} and (A ∪ B) ∩ (A ∪ C) = {1, 2, 3} and hence equal

- A ∩ ( B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Say A = {1,2}, B = {2, 3} and C = {3, 4}

Then A ∩ ( B ∪ C) = {2} and (A ∩ B) ∪ (A ∩ C) = {2} and hence equal

__Disruptive laws for Union and Intersection of Sets__

De-Morgan’s Laws

Let A and B be two subsets of a universal set ξ, then

- (A ∪ B)’ = A’ ∩ B’
- (A ∩ B)’ = A’∪ B’

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

Then A ∪ B = { 1, 2, 3, 4, 5}, therefore (A ∪ B)’ = {6}

A’ = {4, 5, 6} and B’={1, 2, 6}

Therefore A’ ∩ B’ = {6}. Hence proven.

Its very very helpful ……thanks alot..😇😇😍😍😍😍😍😍💋💋💋💋💓👱👰👸👼💏

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