Notesale: Turn your study into money

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Definition of Set :

A set is a well-defined collection of distinct objects, considered as a single entity
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The elements of a set are denoted
using lowercase letters, and the set itself is denoted using an uppercase letter
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The concept of membership refers to
whether an element belongs to a particular set or not
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for example ’ a’ ∈”A” , read as ’a’ belong to set ‘A’ mean element
‘a’ is member of set’ A’
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Representation of Sets:
Sets can be represented in several ways
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Element of the set are separated with coma For example:
The set of natural numbers less than 5 can be represented as: {1, 2, 3, 4}
The set of prime numbers between 10 and 20: {11, 13, 17, 19}
If an element appears more than once in a set, it is still considered a single element
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Set builder notation offers an alternative way to define a set
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For example:
B = {x | x is a prime number less than 20}
C = {x | x is an even number between 10 and 30}

Set operation,
Empty Set:
The set that contains no elements is called the "empty set" or "null set" and is denoted by the symbol
∅ or {}
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Cardinality of a Set:
The cardinality of a set refers to the number of elements it contains
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For example:
If D = {1, 2, 3, 4}, then |D| = 4
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A is said to be a subset of B (denoted as A ⊆ B) if every element of A is also
an element of B
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If there is at least one element in B
that is not in A, then A is called a proper subset of B (denoted as A ⊂ B)
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∅ is also subset of every set
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For the above example P(A) = 4

Universal Set:
The "universal set" is the set that contains all the elements under consideration in a particular context
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Any set under consideration is a subset of the universal set
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A = {1, 2, 3,7}
B = {1, 2, 3, 4, 5}
A ∩ B = {1,2,3,}

Union Set:

The union of A and B (denoted as A ∪ B) is the set of elements that are present in A and B both
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Let U = {1, 2, 3, 4, 5,7, } and A ={1,2,3,7,} A' = { 4,5,}

Difference of set:

Difference of two set A and B is denoted as A- B is a set that contain all element of set A but not in set
B
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Set Relationships:
Disjoint Sets:
Two sets A and B are said to be "disjoint" if their intersection is the empty set, meaning they share no
common elements
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Equal Set:

Two set A and Sets B are equal when they contain the same Elements
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Overlaping Set :

Sets are overlaping when the sets are sharing Elements
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in other words their intersection
set is empty setFor example, let's consider two sets:
Set A = {1, 2, 3}
Set B = {4, 5, 6}
A∩B=∅
It's important to note that if sets are mutually exclusive, their union will be equal to the sum of their
individual sizes because there are no repeated elements between them
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Union Identity Law: A ∪ ∅ = A

The union of set A with the empty set ( ∅ ) results in set A itself because the empty set contains no
elements and does not affect the union
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Then A ∪ ∅ = {1, 2, 3}
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Example: If U = {1, 2, 3, 4, 5} and A = {2, 3}, then A ∩ U = {2, 3}
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Union Domination Law: A ∪ U = U

The union of set A with the universal set (U) results in the universal set U because U contains all
elements, and the union will include all elements of U
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Then A ∪ U = {1, 2, 3, a, b, c}
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Example: If A = {apple, banana, orange} and ∅ denotes the empty set, then A ∩ ∅ = ∅
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Union Idempotent Law: A ∪ A = A

The union of set A with itself gives back set A, as any element present in A will only be counted once
in the union
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Then A ∪ A = {x, y, z}
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Example: If A = {red, green, blue}, then A ∩ A = {red, green, blue}
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Union Complement Law: A ∪ A' = U

The union of set A with its complement (A') results in the universal set U, as A' contains all elements
not present in A, and the union covers all elements
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Then A' = {1, 2, 5} according to union complement law
A ∪ A' = {3, 4,} ∪ {1, 2, 5} = {1, 2, 3, 4, 5} = U

Intersection Complement Law: A ∩ A' = ∅
The intersection of set A with its complement (A') results in the empty set, Considering the above
example, A ∩ A' = { 3, 4,} ∩ { 1,2,5, } = ∅

De Morgan's Laws:

De Morgan's Laws describe the relationship between union and intersection when taking
complements of sets
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For example : Let universal set U = { 1,2,3,4,5,6,7,8,9, } A = { 1,2,5,6, } B = { 2, 3,5,6,7, }
A’ = U - A = { 1,2,3,4,5,6,7,8,9, } - { 1,2,5,6, } = { 3,4,7,8,9, }
B’ = U - B = { 1,2,3,4,5,6,7,8,9, } - { 2,3,5,6,7, } = {1, 4, 8,9, }
A U B = { 1,2,5,6, } U { 2, 3,5,6,7, } = { 1,2,3,5,6,7, } (A ∪ B)' = U - (A ∪ B)
(A ∪ B)' = { 1,2,3,4,5,6,7,8,9, } - { 1, 2, 3,5,6,7, } = {,4,8,9, }
A' ∩ B' = { 3,4,7,8,9, } ∩ { 1, 4, 8,9, } = { 4,8,9, }
As per De Morgan first Law (A ∪ B)' = A' ∩ B' therefore {,4,8,9, } = {,4,8,9, } proved
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