Set Operation


Set and Notation

Set theory is a branch of mathematics. It studies the properties of well-defined collections of objects. It also helps in counting principle. Therefore study of set begins with counting.
A German mathematician Georg Cantor (1845–1918) has conceptualized the modern study of set theory. According to him set is a well-defined collection of objects in which it is possible to determine if a given object is included in the collection.
Set is denoted by a single capital letter (upper cases) of English alphabets such as A, B, C … and so forth. For example

A = {a, e, i, o, u} :The set of vowels

The objects in a set are known as elements or members of the set. The elements in a set are enclosed within middle brackets. For instance, a set A of vowels in English is written within {…} and it is written as
A = {a, e, i, o, u}.

The elements of set are denoted by small letters of English alphabets unless and otherwise stated if applicable. However, the elements in a set can be material objects such as books, pens, alphabets, people etc. or conceptual objects such as numbers, points etc.
We use \( \in \) and \( \notin \) symbols to represent if an element belong to a set or not.

The key relation between sets is membership – when one set is an element of another.
If a is a member of B, this is denoted a ∈ B, while if c is not a member of B then c ∉ B.
For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and C = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 285 ∈ C; but
9 ∉ C and green ∉ B.

Please Note it

Dear learner, please be careful to distinguish the symbols \( 0, \{0\}, \phi, \{\phi\} \)

Question

Is 0 equal to {0} ?




Describing a Set

There are three ways of describing, or specifying the members of, a set.

Semantic definition

The first way is by intentional definition, using a rule or semantic description. For Example

  1. A is the set whose members are the first four positive integers.
  2. B is the set of colors of the French flag.

Extensional definition

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets. For Example

  1. C = {4, 2, 1, 3}
  2. D = {blue, white, red}

In this method, every element of a set must be unique; no two members may be identical. However, the order in which the elements of a set are listed is ignored (unlike for a sequence or tuple).
Combining these two ideas into an example
A={6, 11} , B= {11, 6}
In the examples above, A = B.

Venn-Diagram

The third way is by Venn-Diagram.
A diagram to represent the relationship among the sets is called Venn-diagram. It was named after an English philosopher John Venn (1834-1923). In a Venn-Diagram, rectangular region represent universal set, and other subsets by circular regions. For Example
If U={1,2,3,4,5,6,7,8,9,10}
A={1,2,3,4,5}
B={4,5,6,7,8}
Then, the Venn-Diagram can be described as below.




Cardinality of Set

The concept and notation of Cardinality are due to Georg Cantor who defined the notion of cardinality and realized that sets can have different cardinalities. In summary,

  1. The cardinality of finite set A is n(A)
  2. The cardinality of countable set is ℵ_0 (read as aleph-naught or aleph-zero or aleph-null)
  3. The cardinality of uncountable set is 𝑐 (read as continuum)

The cardinality of a set A is the number of elements of the set A . The cardinality of a set A is usually denoted by n(A) but it can also be denoted as Card(A). For example:

  • If \( A = \{x: x< 4, x \in W \}\) then A = {0, 1, 2, 3} and n (A) = 4
  • If B = { letters in the word “mathematics”} then B = {m, a, t, h, e, i, c, s} and n(B) = 8.



Types of Set

In this section, we discuss about common types of sets

Empty set

A set having no member (element) is called empty or null or void set and it denoted by \( \{ \} \) or \( \phi\) (Phi). For example

  1. \( A = \{ x: x \text{ is a natural number less than } 1\} = \phi \)
  2. \( B = \{ x: 5 < x < 6, x \in N\} = \phi \)
  3. \( C = \{ \text{ boy student in a girl college} \} = \phi \)

Note: If \( A = \phi \) then \( n (A) = 0\) but if \( A = \{ \phi \} \) then \( n (A) = 1\)

Singleton set

A set having only one element is called singleton set. For example

  1. \( A= \{ \text{ the highest pick of the world } \} = \{ \text{ Mt. Everest} \} \)
  2. \( B = \{ x: x < 1, x\in W\} = \{ 0\} \)
  3. \( C = \{ x: x2 = 9, x\in N\} = \{ 3\} \)

Doubleton set

A set having only two elements is called doubleton set. For example

  1. \( P = \{ \text{ faces of a coin} \} = \{ H, T\} \)
  2. \( Q = \{ x: x^2 = 1, x\in Z\} = \{ -1, 1\} \)
  3. \( R = \{ a, \{ b, c\} \} \)

Finite set

A set having predetermined finite number of elements is called finite set. For example
  • \( A = \{ \text{ whole numbers less than} 10\} = \{ 0,1,2,3,4,5,6,7,8,9\} \)
  • \( B = \{ 1, 2, 3, …., 99, 100\} \)
  • \( C= \{ \text{ set of English alphabets} \} \)

NOTE

A set is called finite if it has a cardinality n for some \( n \in N \). Some examples of finite set are

  • \( \phi\) and singleton sets are finite
  • The union of two finite sets is finite
  • Subset of a finite set is finite

Infinite set

A set containing never-ending elements is called infinite set. For example

  • \( A = \{ x: x \in N, x > 1\} = \{ 2, 3, 4, , ...\} \)
  • \( B = \{ x: x \in W, x = 2n\} = \{ 0, 2, 4, 6, ...\} \)
  • \( C=\{ x:x\in Q\} \)

A set is infinite, if it is not finite. Some examples of infinite set are

  • The set \( N\) of natural numbers is an infinite set
  • Superset of infinite set is infinite

Denumerable sets

An infinite set is denumerable if it is equivalent to the set of natural numbers \( N\) . More precisely, a set \( A\) is said to be a denumerable if there exists a bijection \( f: A\to N\) . The cardinality of denumerable set is denoted by the Hebrew alphabet ‘aleph null’ \( \aleph_0\) .Some example of denumerable sets are as below.

  • The set \( N\) of natural numbers
  • The set \( Z\) of integers
  • The set \( P\) of prime numbers
  • The set \( O\) of odd integers
  • The set \( E\) of even integers
  • The set \( Q\) of rational numbers
  • The set \( N \times N\) of natural numbers
  • The union of a denumerable set and a singleton set is denumerable
  • The union of two denumerable sets is denumerable.
  • Union of infinite denumerable sets is denumerable.

Some examples of non-denumerable sets are as follows.

  • The set \( R\) of all real numbers is non-denumerable
  • The set \( R/Q\) of all irrational numbers is non-denumerable

Countable and Uncountable set

A set \( A\) is called countable if it is finite or denumerable. Otherwise, the set \( A\) is called uncountable. The cardinality of countable set is either \( n\) or \( \aleph_0\) . The cardinality of uncountable set is \( c\) . Some example of countably infinite sets are as follows

  • The set \( E\) of even numbers
  • The set \( P\) of prime numbers
  • The set \( Z\) of integer numbers
  • The set \( Q\) of rational number
  • Any finite set is countable
  • Every subset of a countable set is countable
  • Union of countable sets countable
  • The Cartesian product finitely many countable sets is countable.
  • A subset of a countable set is countable.
  • The set \( R\) of real numbers.
  • The set \( R/Q\) of irrational numbers.
  • A superset of an uncountable set is uncountable.



Relation Between Sets

In mathematics, a relation between sets is a subset of their Cartesian product. However, we can also define relation as follows.

Subset

A set \( A\) is said to be a subset of another set\( B\) if every element of \( A\) is also an element of the set\( B\) . If \( A\) is subset of\( B\) , then it is written as \(A \subset B\) and read as “\( A\) is contained in\( B\) ” or “\( A\) is a subset of\( B\) ”. For example:
Let \(A = \{1, 2, 3\},B = \{3, 4, 5,6\}\) and \(C = \{1, 2, 3, 4, 5\}\) then \(A \subset C\) but \(B \not\subset C\).
In usual notation of set of numbers, the relation between them are as below \(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset\mathbb{Q} \subset \mathbb{R} \).
There are two types of subsets

  1. Proper subset
    If \( A \subset B\) and \( A \ne B\) then \( A\) is called a proper subset of\( B\) . In this case,\( B\) is superset of \( A\)
  2. Improper subset
    If \( A \subset B\) and \( A = B\) then \( A\) is called an improper subset of\( B\) . It is written as \( A \subseteq B\) .
    Note:
    • The empty set \( \phi \) is a subset of every set.
    • Every set is a subset of itself.
    • Every non-empty set has at least two subsets
    • The total number of possible subsets of a set with n-elements is \( 2^n\) .

Power Set

Let \( S\) is a set. Then the set of all the possible subsets of \( S\) is called power set of \( S\) . It is denoted by \( P(S)\) . For example,
if \( S = \{a, b, c\}\) then \( P(S) = \{\phi, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\},\{a, c\}, \{a, b, c\}\}\) .

Thus, the power set of a set S is the set of all subsets of S, including S itself and the empty set. For example, the power set of the set {1, 2, 3} is
{{1, 2,3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}.
The power set of a set S usually written as P(S).

The power set of a finite set with n elements has 2n elements.
This relationship is one of the reasons for the terminology power set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains
23 = 8 elements.
The power set of an infinite (either countable or uncountable) set is always uncountable.

  • If \( P (S)\) is the power set of a set \( S\) then \( n (P(S)) = 2^{n(S)}\) .
  • Power set of a finite set is finite.
  • \( S \in P(S)\) , that is, Set \( S\) is an element of power set of \( S\) .

Equal Sets

Two or more sets are called equal (or identical or same) if they consist same elements. For example,
if \( A = \{1, 2\}\) and \( B = \{2, 1\}\) then \( A = B\) but the sets \( C = \{12\}\) and \( D =\{21\}\) are not equal.

Equivalent Sets

Two sets \( A\) and\( B\) are called equivalent if their cardinal number is same, i.e., \( n (A) = n (B)\) . The symbol to denote equivalent sets is “\( \sim\) ”. For example,
if \( A = \{1, 2, 3\}\) and \( B = \{p, q, r\}\) then \( A \sim \) B.

Overlapping Sets

Two sets \( A\) and\( B\) are called overlapping set if they do have some common element. For example,
if \( A = \{1,2,3\}\) and \( B = \{3,4,5\}\) then \( A\) and\( B\) are overlapping sets as \( \{3\}\) is common to both sets \( A\) and\( B\) .

Disjoint Sets

Two sets \( A\) and \( B\) are called disjoint if they have no elements in common. For example, if \( A = \{1, 2, 3\}\) and \( B = \{4,5, 6\}\) then \( A\) and\( B\) are disjoint as they have no element in common.

Comparable Sets

Two sets \( A\) and\( B\) are said to be comparable if \( A \subset B \) or \( B \subset A\) . For example, the sets \( A=\{a, b, c\}\) , and \( C=\{a, b, c, d\}\) are comparable. But the sets \( C=\{a, b, c\}\) and \( D=\{a, c, d, e\}\) are not comparable (incomparable) sets.




Set Operation

In real number system, we can do four fundamental operation to form new number by combining or manipulating one or more existing numbers. For example, given two numbers \(2\) and \(3\) , we can use

  1. \(+\) to form a new number \(5\) by \(2+3\)
  2. \(\times\) to form a new number \(6\) by \(2 \times 3\)
  3. we can do Set operation to form new Set by combining or manipulating one or more existing Sets.
  4. Set operation helps to combine two or more sets together to form a new set.
  5. The common example of set operations are: Union, Intersection, Difference, and Complement

Union of Sets

Let A and B be any two sets. Then union of sets A and B is a new set consisting all the elements of A and B without repetition. The union is the smallest set containing elements of A and B.
It is denoted by AUB and read as “A union B” or “A cup B”.
Mathematically,
AUB = {x: x ∈ A or x ∈ B}.

मानौ A र B कुनै दुई समुहहरू छन । अब समुह A र B को संयोजन (union) भनेको एउटा नयाँ समुह हो जुन A र B का सबै सदस्यहरु समावेश भई बनेको हुन्छ। संयोजन समुह A र B बाट बन्ने सबैभन्दा सानो समुह हो । यसलाई AUB ले जनाईन्छ र "A संयोजन B" भनेर पढिन्छ।

गणितिय भाषामा,
AUB = {x: x ∈ A or x ∈ B}.

Example 1

If A={ 1,2,3,4,5} and B={4,5,6,7,8}, then find A∪B
Solution
In this example, A={ 1,2,3,4,5} and B={4,5,6,7,8}
Thus,
A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
or A∪B={4,5} ∪{1,2,3}∪{6,7,8}
or A∪B={1,2,3,4,5,6,7,8}

the shaded region is A∪B

Example 2

If A={ 1,2,3} and B={6,7,8}, then find A∪B
Solution
In this example, A={ 1,2,3} and B={6,7,8}
Thus,
A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
or A∪B={ }∪{1,2,3}∪{6,7,8}
or A∪B={1,2,3,6,7,8}

the shaded region is A∪B

Example 3

If A={ 1,2,3,4,5} and B={4,5}, then find A∪B
Solution
In this example, A={1,2,3,4,5} and B={4,5}
Thus,
A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
or A∪B={4,5} ∪{1,2,3}∪{}
or A∪B={1,2,3,4,5}

the shaded region is A∪B

Example 4

If B={ 1,2,3,4,5} and A={4,5}, then find A∪B
Solution
In this example, B={1,2,3,4,5} and A={4,5}
Thus,
A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
or A∪B={4,5} ∪{1,2,3}∪{}
or A∪B={1,2,3,4,5}

the shaded region is A∪B

Example 5

If A={1,2,3,4,5} and B={1,2,3,4,5}, then find A∪B
Solution
In this example, A={1,2,3,4,5} and B={1,2,3,4,5}
Thus,
A∪B={Common Elements of A and B} ∪ {Remaining element of A} ∪ {Remaining element of B}
or A∪B={1,2,3,4,5} ∪{}∪{}
or A∪B={1,2,3,4,5}




Algebra of Sets

Properties of Union on Sets

  • A U B = B U A
  • A U A= A
  • A U Φ = A
  • A U U = U
  • If A ⊂ B then A U B = B
  • ( A U B ) U C = A U (B U C)

Properties of Intersection on Sets

  • A ∩ B= B ∩ A
  • A ∩ A= A
  • A ∩ Φ = Φ
  • A ∩ U = A
  • (A ∩ B) ∩ C= A ∩ (B ∩ C)
  • If A ⊂ B then A ∩ B = A

Properties of Complement on Sets

  • A U A' = U
  • A ∩ A' = Φ
  • U' = Φ
  • Φ ' =U

Algebra of sets

The algebra of sets develops and describes the basic properties and laws on sets. The common law on set-theoretical operations of union, intersection, complementation and other relations are mentioned below.

Properties on Set Equality and Set Inclusion

Let A, B and C be subsets of a universal set U then

  • A = B ⇒ B = A
  • A = B, B = C ⇒ A = C
  • A ⊂ B , B ⊂ C ⇒ A ⊂ C
  • A ⊂ Φ ⇒ A = Φ

Laws of algebra on sets

Let A, B and C be subsets of a universal set U then

Laws Identities over union Identities over Intersection
Idempotent Laws A U A = A A ∩ A = A
Identities Laws A U U = U and A U Φ = A A ∩ U = A and A ∩ Φ = Φ
Complement Laws AU A' = U and (A')'=A A ∩ A' = Φ and U'=Φ
Commutative Laws A U B = B U A A ∩ B = B ∩ A
Associative Laws ( A U B )U C = AU (BU A) (A∩ B)∩ C=A∩ (B∩ A)
Distributive Laws AU (B∩ C)=( A U B )∩ (AU C) A∩ (BU C)=(A∩ B)U (A∩ C)
De-Morgan’s Laws (AU B)'=A' ∩ B' (AU B)'=A' ∩ B'

Theorem 1

Prove that (AUB)'=A'∩B'

Set Builder Method

We start by LHS, then

(AUB)' =U-(AUB)
={x:x∈U and x∉(AUB)}
={x:x∈U and (x∉ A and x∉B)}
={x: (x∈U and x∉ A) and (x∈U and x∉B)}
={x:x∈A' and x∈B'}
=A'∩B'

Thus
(AUB)'=A'∩B'

Membership Tabular Method

A B AUB (AUB)' A' B' A'∩B'
1 1 1 0 0 0 0
1 0 1 0 0 1 0
0 1 1 0 1 0 0
0 0 0 1 1 1 1

Thus
(AUB)'=A'∩B'

Venn-Diagram Method

the shaded region is (AUB)'

the shaded region is B'

the shaded region is B'

Thus
(AUB)'=A'∩B'




दुईवटा समुहहरु सम्मिलित भएको शाब्दिक समस्याहरु समाधान गर्ने योजना

अंकगणितमा +,-,x, ÷ लाई चारवटा आधारभतु क्रियाहरु मानिन्छ । In school mathematics, यि चारवटा आधारभतु क्रियाहरु बाहेक "Set" लाई पनि counting principle को रुपमा आधारभुत क्रिया भनिन्छ । भेनडायग्राम को प्रयोग गरेर समुहबाट counting system सम्बन्धी समस्याहरुको समाधान गर्न सकिन्छ । जसलाई big ideas पनि भनिन्छ।

दुईवटा समुहहरु सम्मिलित भएको शाब्दिक समस्याहरु समाधान गर्ने योजना
  1. Testing the set operation
  2. Testing the Cardinality
  3. Parts of the sets (4 disjoint pieces and 16 possibilities)
  4. Arithmetic’s on Cardinality
  5. Identification of set notation
  6. Problem solving (Context specific)



Parts of the sets

Below is a Venn diagram involving two sets A and B




We can make four disjoint parts of the above Venn-diagram, which are as below.

  1. Part 1: A-B

    This part is known as A difference B
    It is denoted by A−B
    It is also denoted by Ao
    It represents the cardinality (or elements) which lies in only A but not in B

  2. Part 2: A∩B

    This part is also known as A intersection B
    This part is denoted by A∩B
    This parts represents the cardinality (or elements) which lies both in A and B.

  3. Part 3: B-A

    This part is also known as B difference A
    This part is denoted by B-A
    This part is also denoted by Bo
    This parts represents the cardinality (or elements) which lies in only B but not in A

  4. Part 4: (AUB)'

    This part is also known complement of A union B
    This parts represents the cardinality (or elements) which lies neither in A nor in B




Different notion of sets using four disjoint parts

Below is a Venn diagram involving two sets A and B

Here are four disjoint parts of the Venn-diagram. These four parts are

  • A0 (or A-B)[red color]
  • B0 (or B-A)[green Color]
  • A∩B [gray Color]
  • (AUB)' [pink color]



Now,how many different sets can be formed using these four disjoint parts.
Using these four disjoint parts, all together 16 different set notation can be formed. More explicitely

  1. 1 set notation can be formed taking 0 parts out of 4 disjoint parts
  2. four different set notation can be formed taking 1 parts out of 4 disjoint parts
  3. six different set notation can be formed taking 2 parts out of 4 disjoint parts
  4. four different set notation can be formed taking 3 parts out of 4 disjoint parts
  5. 1 set notation can be formed taking 4 parts out of 4 disjoint parts
Possible CombinationsPossible number of sets Set Notations
Set with zero parts1 \( \phi \)
Set with one parts4 \(A_0,B_0,A \cap B, (A \cup B)' \)
Set with two parts6 \( A,B,A',B,A \triangle B, (A \triangle B)'\)
Set with three parts4 \( (A-B)', (B-A)',(A \cap B)', A \cup B \)
Set with four parts1 \( U \)

अब, माथिको भेन चित्रको आधारमा चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर कतिवटा फरक फरक समुहहरु बनाउन सकिन्छ?
माथिको चारवटा अलगिएका समुहहरुलाई प्रयोग गरेर जम्मा 16 वटा फरक फरक समुहहरु बनाउन सकिन्छ । जसमा

  1. 0 वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।
  2. १ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
  3. २ वटा भागलाई प्रयोग गरेर ६ वटा समुह बनाउन सकिन्छ।
  4. ३ वटा भागलाई प्रयोग गरेर ४ वटा समुह बनाउन सकिन्छ।
  5. ४ वटा भागलाई प्रयोग गरेर १ वटा समुह बनाउन सकिन्छ।

Therefore, all together 16 different set notation can be formed.
These 16 different set notation are given below.




  1. Part 1: 𝜙

    Set Notation:𝜙
    This part is formed taking 0 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as empty set.
    It contains no cardinality (or elements) of the sets A or B or U.
    समुह संकेत :𝜙
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये 0 वटा भाग को प्रयोग भएको छ।
    यसलाई खाली समुह पनि भनिन्छ ।
    यस समुहमा समुह A वा B वा U कुनैमा पनि नभएका सदस्यहरु पर्दछन।


  2. Part 2: A-B

    Set Notation:A-B (or A0)
    This part is formed taking 1 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as A difference B.
    It contains the cardinality (or elements) that belong to set A only, but not in B
    समुह संकेत :A-B (or A0)
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को फरक समुह भनिन्छ ।
    यसमा समुह A मा भएका तर B मा नभएका सदस्यहरु पर्दछन।

  3. Part 3: B-A

    Set Notation:B-A (or B0)
    This part is formed taking 1 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as B difference A.
    It contains the cardinality (or elements) that belong to set B only, but not in A
    समुह संकेत :B-A (or B0)
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को फरक समुह भनिन्छ ।
    यसमा समुह B मा भएका तर A मा नभएका सदस्यहरु पर्दछन।

  4. Part 4: A∩B

    Set Notation:A∩B
    This part is formed taking 1 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as A intersection B.
    It contains the cardinality (or elements) that belong to sets A and B both
    समुह संकेत :A∩B
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को प्रतिच्छेदन समुह भनिन्छ ।
    यसमा समुह A र B दुवैमा पर्ने साझा सदस्यहरु पर्दछन।

  5. Part 5: (AUB)'

    Set Notation: (AUB)'
    This part is formed taking 1 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of A union B.
    It contains the cardinality (or elements) that belong to sets Neither A nor B nor both
    समुह संकेत : (AUB)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये १ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को संयोजन को पुरक समुह भनिन्छ ।
    यसमा समुह A वा B दुवैमा नपर्ने सदस्यहरु पर्दछन।


  6. Part 6: A

    Set Notation: A
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as A .
    It contains the cardinality (or elements) that belong to sets A
    समुह संकेत :A
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई A समुह भनिन्छ ।
    यसमा समुह Aमा पर्ने सबै सदस्यहरु पर्दछन।

  7. Part 7: B

    Set Notation:B
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as B .
    It contains the cardinality (or elements) that belong to sets B
    समुह संकेत :B
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई B समुह भनिन्छ ।
    यसमा समुह B मा पर्ने सबै सदस्यहरु पर्दछन।

  8. Part 8: A'

    Set Notation:A'
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of A .
    It contains the cardinality (or elements) that does NOT belong to set A
    समुह संकेत :A'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई A को पुरक समुह भनिन्छ ।
    यसमा समुह A मा नपर्ने सबै सदस्यहरु पर्दछन।

  9. Part 9: B'

    Set Notation:B'
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of B .
    It contains the cardinality (or elements) that does NOT belong to set B
    समुह संकेत :B'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई B को पुरक समुह भनिन्छ ।
    यसमा समुह B मा नपर्ने सबै सदस्यहरु पर्दछन।

  10. Part 10: A∆B

    Set Notation:A∆B
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that belong to only one set
    समुह संकेत :A∆B
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजन समुह भनिन्छ ।
    यसमा समुह (A-B) वा (B-A) मा पर्ने सबै सदस्यहरु पर्दछन।

  11. Part 11: (A∆B)'

    Set Notation: (A∆B)'
    This part is formed taking 2 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of the union of (A-B) and (B-A) .
    It contains the cardinality (or elements) that does NOT belong to only one set
    समुह संकेत : (A∆B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये २ वटा भाग को प्रयोग भएको छ।
    यसलाई (A-B) र (B-A) को संयोजनको पुरक समुह भनिन्छ ।
    यसमा समुह (A-B) र (B-A) कुनैमा पनि नपर्ने सदस्यहरु पर्दछन।


  12. Part 12: AUB

    Set Notation: AUB
    This part is formed taking 3 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as union of A and B.
    It contains the cardinality (or elements) that belongs to either A or B or Both
    समुह संकेत : AUB
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को संयोजन समुह भनिन्छ ।
    यसमा समुह A वा B मा पर्ने सदस्यहरु पर्दछन।

  13. Part 13: \(A_0'\)

    Set Notation: (A0)'
    This part is formed taking 3 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of A difference B.
    It contains the cardinality (or elements) that does Not belong to A only
    समुह संकेत : (A0)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई A र B को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

  14. Part 14: \(B_0'\)

    Set Notation: (B0)'
    This part is formed taking 3 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of B difference A.
    It contains the cardinality (or elements) that does Not belong to B only
    समुह संकेत : (B0)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को फरक को पुरक समुह भनिन्छ ।
    यसमा समुह B र A को फरकमा नपर्ने सबै सदस्यहरु पर्दछन।

  15. Part 15: (A∩B)'

    Set Notation: (A∩B)'
    This part is formed taking 3 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as complement of A intersection B.
    It contains the cardinality (or elements) that does Not belong to both A and B
    समुह संकेत : (A∩B)'
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये ३ वटा भाग को प्रयोग भएको छ।
    यसलाई B र A को प्रतिच्छेदन को पुरक समुह भनिन्छ ।
    यसमा समुह A र B को प्रतिच्छेदनमा नपर्ने सबै सदस्यहरु पर्दछन।

  16. Part 16: U

    Set Notation:U
    This part is formed taking 4 parts out of the four parts A0, B0,A∩B and (AUB)'
    This part is also known as full (Universal) set.
    It contains all cardinality (or elements) of the sets A or B or U.
    समुह संकेत :U
    यो समुह बन्न को लागी दिएका चारवटा भागहरु A0, B0,A∩B र (AUB)'मध्ये ४ वटा भाग को प्रयोग भएको छ।
    यसलाई सर्वव्यापक समुह भनिन्छ ।
    यसमा समुह A वा B वा U भएका सबै सदस्यहरु पर्दछन।




Arithemetic of Cardinality

Arithmetic of cardinality in sets refers to the mathematical operations that involve counting the number of elements (cardinality) within sets. When solving verbal problems involving sets, you might encounter situations where you need to perform arithmetic operations such as addition, subtraction, multiplication, and division on the cardinalities of sets to find the desired information.
Here are some common scenarios where arithmetic of cardinality comes into play when solving verbal problems related to sets:

  1. Union of Sets: When we need to find the total number of elements in the union of two or more sets, we use the concept of cardinality. For example, if we have sets A and B, the cardinality of their union (A ∪ B) can be calculated by adding the cardinalities of A and B and then subtracting the cardinality of their intersection (A ∩ B) to avoid double counting any shared elements.
    n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
  2. Complements: The complement of a set A with respect to a larger set U (universal set) contains all elements in U that are not in A. You can calculate the cardinality of the complement by subtracting the cardinality of A from the cardinality of U.
    n(A') = n(U) - n(A)
  3. Subtraction of Sets: When you want to find the number of elements in one set that are not in another set, you can use subtraction of cardinalities. For instance, if you have sets A and B, the cardinality of the difference A - B is found by subtracting the cardinality of B from the cardinality of A.
    n(A - B) = n(A) - n(B)



    1. Test your Understandings

      From the Venn-diagram given below, find the cardinal number given sets.

      1. \(n(A \triangle B) \)

      2. \(n(A-B) \)

      3. \(n(B-A) \)

      4. \(n(A \cap B) \)

      5. \(n(A \cup B)' \)

      6. \(n(A \cup B) \)

      7. \(n(A) \)

      8. \(n(B) \)

      9. \(n(A)' \)

      10. \(n(B)' \)

      If you want more practice, refresh the page

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