General Mathematis with the Topic of SETs Story

GENERAL MATHEMATICS
SET THEORY
STATE COLLEGE OF EDUCATION
MIRPUR

INTRODUCTION TO SETS
A set is a well-defined collection of distinct objects. Well-defined collection means
that there exists a rule with the help of which it is possible to tell whether a given
object belongs or does not belong to given collection. Generally, sets are denoted by
capital letters A, B, C, X, Y, Z etc.
Sets are fundamental objects in mathematics used to define relationships between
elements.
Sets can be finite or infinite, depending on the number of elements they contain.

ELEMENTS OF A SET
The objects or elements in a set can be anything, such as numbers, letters, or even
other sets.
Elements of a set are denoted by listing them within curly braces, separated by
commas.
Each element in a set is unique, and the order of elements does not matter.

Symbols Meaning
{ } Symbol of set
U Universal set
n(X) Cardinal number of set X
b ∈ A 'b' is an element of set A
a ∉ B 'a' is not an element of set B
∅ Null or empty set
A U B Set A union set B
A ∩ B Set A intersection set B
A ⊆ B Set A is a subset of set B
B ⊇ A Set B is the superset of set A
SETS SYMBOLS

TYPES OF SET
We have several types of sets in Math's. They are empty set, finite and
infinite sets, proper set, equal sets, etc. Let us go through the classification
of sets here.
Empty Set:
A set which does not contain any element is called an empty set or void
set or null set. It is denoted by { } or Ø.
A set of apples in the basket of grapes is an example of an empty set
because in a grapes basket there are no apples present.
Singleton Set:
A set which contains a single element is called a singleton set.
Example: There is only one apple in a basket of grapes.

Finite set:
A set which consists of a definite number of elements is called a finite set.
Example: A set of natural numbers up to 10.
A = {1,2,3,4,5,6,7,8,9,10}
Infinite set:
A set which is not finite is called an infinite set.
Example: A set of all natural numbers.
A = {1,2,3,4,5,6,7,8,9……}
TYPES OF SET

TYPES OF SET
Equivalent Set:
If the number of elements is the same for two different sets, then they are called
equivalent sets. The order of sets does not matter here. It is represented as:
n(A) = n(B)
where A and B are two different sets with the same number of elements.
Example:
If A = {1,2,3,4} and B = {Red, Blue, Green, Black}
In set A, there are four elements and in set B also there are four elements.
Therefore, set A and set B are equivalent.

Equal sets:
The two sets A and B are said to be equal if they have exactly the same elements, the
order of elements do not matter.
Example: A = {1,2,3,4} and B = {4,3,2,1}
A = B
Disjoint Sets:
The two sets A and B are said to be disjoint if the set does not contain any common
element.
Example: Set A = {1,2,3,4} and set B = {5,6,7,8} are disjoint sets, because there is no
common element between them.
TYPES OF SET

TYPES OF SET
Subsets:
A set ‘A’ is said to be a subset of B if every element of A is also an element of B,
denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In
general, a subset is a part of another set.
Example: A = {1,2,3}
Then {1,2} ⊆ A.
Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.
Note: The set is also a subset of itself.
If A is not a subset of B, then it is denoted as A⊄B.

TYPES OF SET
Proper Subset:
If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.
Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B =
{2,5,7}
But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.
Superset:
Set A is said to be the superset of B if all the elements of set B are the elements of set A.
It is represented as A ⊃ B.
For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.

TYPES OF SET
Universal Set:
A set which contains all the sets relevant to a certain condition is called the
universal set. It is the set of all possible values.
Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:
U = {1,2,3,4,5}

OPERATIONS ON SETS
In set theory, the operations of the sets are carried when two or more sets
combine to form a single set under some of the given conditions. The basic
operations on sets are:
 Union of sets
 Intersection of sets
 A complement of a set
 Set difference
 Cartesian product of sets.
Basically, we work more on union and intersection of sets operations,
using Venn diagrams.

Union of Sets
If set A and set B are two sets, then A union B is the set that contains all the elements
of set A and set B. It is denoted as A ∪ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:
A ∪ B = {1,2,3,4,5,6}
OPERATIONS ON SETS

Intersection of Sets
If set A and set B are two sets, then A intersection B is the set that contains only the
common elements between set A and set B. It is denoted as A ∩ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:
A ∩ B = { } or Ø
Since A and B do not have any elements in common, so their intersection will give
null set.
OPERATIONS ON SETS

OPERATIONS ON SETS
Complement of Sets
The complement of any set, say P, is the set of all elements in the universal set that
are not in set P. It is denoted by P’
Properties of Complement sets
 P ∪ P′ = U
 P ∩ P′ = Φ
 Law of double complement : (P′ )′ = P
 Laws of empty/null set(Φ) and universal set(U), Φ′ = U and U′ = Φ.

OPERATIONS ON SETS
Cartesian Product of Sets
If set A and set B are two sets then the Cartesian product of set A and set B is a set
of all ordered pairs (a,b), such that a is an element of A and b is an element of B.
It is denoted by A × B.
We can represent it in set-builder form, such as:
A × B = {(a, b) : a ∈ A and b ∈ B}
Example: set A = {1,2,3} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

Difference of Sets
If set A and set B are two sets, then set A difference set B is a set which
has elements of A but no elements of B. It is denoted as A – B.
Example: A = {1,2,3} and B = {2,3,4}
A – B = {1}
OPERATIONS ON SETS

REPRESENTATION OF SET
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or
{Bat, Ball, Wickets}.
 Statement or Descriptive form
 Roster or Tabular Form
 Set Builder Form

Statement or Descriptive Form:
In statement form, the well-defined descriptions of a member of a set are
written and enclosed in the curly brackets.
For example, the set of even numbers less than 15.
In statement form, it can be written as {even numbers less than 15}.

Roster or Tabular Form:
In Roster form, all the elements of a set are listed.
For example, the set of natural numbers less than 5.
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….
Natural Number less than 5 = 1, 2, 3, 4
Therefore, the set is N = { 1, 2, 3, 4 }

Set Builder Form
The general form is, A = { x : property }
Example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}
Also, Venn Diagrams are the simple and best way for visualized representation of
sets.

Order of Set:
The order of a set defines the number of elements a set is having. It
describes the size of a set. The order of set is also known as
the cardinality.
The size of set whether it is is a finite set or an infinite set, said to be set of
finite order or infinite order, respectively.
The cardinality of the empty set is zero.
ORDER OF SET

Power Set
A power set is set of all subsets, empty set and the original set itself.
For example, power set of A = {1, 2} is
P(A) = {{}, {1}, {2}, {1, 2}}.
POWER SET

𝑵: the set of all natural numbers = {1, 2, 3, 4, .....}
𝑾: the set of all whole numbers={0,1, 2, 3, 4, .....}
𝒁: the set of all integers = {0, ,±1 ,±2 ,±3, ± 4,…..}
𝐙+ : the set of all positive integers= {1, 2, 3, 4, .....}
𝐙 − : the set of all negative integers = {.....,-4,-3, -2, -1}
SOME IMPORTANT SETS

𝑬: the set of all even numbers = {0,±2, ,±4 ,±6 ,±8,……}
𝑬+:the set of all positive even numbers = {2,4,6,8,..……}
𝑶:the set of all odd numbers = {±1, ±3,±5,±7,..……}
𝑶+
:the set of all positive odd numbers = {1,3,5,7,..……}
𝑶−
:the set of all negative odd numbers = {-1,-3,-5,-7,..……}
SOME IMPORTANT SETS

SOME IMPORTANT SETS
𝑸: set of all rational numbers = {𝑥 | 𝑥 =
𝑝
𝑞
∧ 𝑝 ∈ 𝑍, 𝑞 ∈ 𝑍, 𝑞 ≠ 0}
𝑸′: set of all irrational numbers = {𝑥 | 𝑥 ∈ 𝑅, 𝑥 ∉ 𝑄}
R: set of all real numbers = 𝑸 ∪ 𝑸′

APPLICATIONS OF SETS
Sets are used in various branches of
mathematics, including algebra, calculus,
and probability.
Set theory is foundational in defining
mathematical structures and relationships.
Sets are also used in computer science,
databases, and logic for data organization
and analysis.

CONCLUSION
Sets are essential mathematical structures that help define relationships
between elements.
Understanding sets and their properties is crucial in various fields of
mathematics and beyond.
Explore further applications and challenges in set theory to deepen your
understanding of this fundamental concept.

 A set Q= {𝑥 | 𝑥 =
𝑝
𝑞
∧ 𝑝 ∈ 𝑍, 𝑞 ∈ 𝑍, 𝑞 ≠ 0}
is called a set of
A. Whole numbers
B. Natural numbers
C. Irrational numbers
D. Rational numbers
MCQs

 The different number of ways to describe a set are
A. 1
B. 2
C. 3
D. 4
MCQs

 A set with no element is called
A. Subset
B. Empty set
C. Singleton set
D. Super set
MCQs

 The set 𝑥/𝑥 ∈ 𝑊 ∧ 𝑥 ≤ 101 is
A. Infinite set
B. Subset
C. Null set
D. Finite set
MCQs

 The set having only one element is called:
A. Null set
B. Power set
C. Singleton set
D. Subset
MCQs

 Power set of an empty set is:
A. ∅
B. {𝑎}
C. {∅,{𝑎}}
D. {∅}
MCQs

 The number of elements in power set {1, 2,3} is:
A. 4
B. 6
C. 8
D. 9
MCQs

 If A⊆ 𝐵 then A∪ 𝐵 is equal to:
A. A
B. B
C. ∅
D. None of these
MCQs

 If A⊆ 𝐵 then A∩ 𝐵 is equal to:
A. A
B. B
C. ∅
D. None of these
MCQs

 If A⊆ 𝐵 then A−𝐵 is equal to:
A. A
B. B
C. ∅
D. None of these
MCQs

 ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 is equal to:
A. 𝐴 ∩ (𝐵 ∪ 𝐶)
B. (𝐴 ∪ 𝐵) ∩ 𝐶
C. 𝐴 ∪ 𝐵 ∪ 𝐶
D. 𝐴 ∩ (𝐵 ∩ 𝐶)
MCQs

 𝐴 ∪ (𝐵 ∩ 𝐶) is equal to:
A. 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶
B. 𝐴 ∩ (𝐵 ∩ 𝐶)
C. 𝐴 ∩ 𝐵 ∩ 𝐴 ∩ 𝐶
D. 𝐴 ∪ (𝐵 ∪ 𝐶)
MCQs

 If A and B are disjoint sets, then 𝐴 ∪ 𝐵 is equal to:
A. A
B. B
C. ∅
D. B ∪A
MCQs

 If number of elements in set A is 3 and in set B is 4, then
number of elements in A×B is:
A. 3
B. 4
C. 12
D. 7
MCQs

 If number of elements in se A is 3 and in set B is 2,
then number of binary relations in A×B is:
A. 28
B. 26
MCQs

 If 𝐴 ∪ 𝐵 = ∅ , then set A and B are
A. Sub set
B. Over lapping set
C. Disjoint set
D. Power set
MCQs

 The complement of U is:
A. U
B. ∅
C. Impossible
D. union
MCQs

 The complement of ∅ is:
A. U
B. ∅
C. Impossible
D. union
MCQs

 𝐴 ∪ 𝐴𝑐 = ……
A. U
B. A
C. 𝐴𝑐
D. ∅
MCQs

 𝐴 ∩ 𝐴𝑐 = ……
A. U
B. A
C. 𝐴𝑐
D. ∅
MCQs

 The point (−5 , −7) lies in quadrant.
A. I
B. II
C. III
D. IV
MCQs

 y co-ordinate of every point on x–axis is:
A. +ve
B. −ve
C. Zero
D. 1
MCQs

 x co-ordinate of every point on x–axis is:
A. +ve
B. −ve
C. Zero
D. 1
MCQs

 The domain of {(a,b), (b,c), (c,d)} is:
A. {a,b,c}
B. {b,c,d}
C. {a,b}
D. {a, b,c,d,}
MCQs

 Venn diagram was first used by:
A. John Venn
B. Newton
C. Arthur Clayey
D. John Napie
MCQs

 A subset of A x A is called in A.
A. Set
B. Relation
C. Function
D. Into function
MCQs

 The relation {(a,b),(b,c),(a,d)} is:
A. A function
B. Not a function
C. Range
D. Domain
MCQs

 By definition, which of the following is a set?
A. {a,b,c,a}
B. {1,2,3,2}
C. {l,m,n,o}
D. {0,1,2,3,1}
MCQs

 Which of the following is true?
A. W⊆N
B. Z ⊆ W
C. N ⊆ P
D. P ⊆ W
MCQs

A. P ⊆ N ⊆ Z ⊆ W
B. P ⊆ N ⊆ W ⊆ Z
C. P ⊆ W ⊆ N ⊆ Z
D. P ⊆ Z ⊆ N ⊆ W
MCQs

A. N and W ⊆ Z
B. P and O ⊆ W
C. O and E ⊆ W
D. P and E ⊆ N
MCQs

 N ∩W =.........
A. ∅
B. {0}
C. N
D. W
MCQs

 N ∪W =.........
A. ∅
B. {0}
C. N
D. W
MCQs

 W −N =.........
A. (∅
B. {0}
C. N
D. W
MCQs

 Two sets having no common element are called …… sets
A. Subset
B. Overlapping
C. Disjoint
D. Super
MCQs

 If two sets have some elements common but not all
are called….sets.
A. Sub
B. Overlapping
C. Disjoint
D. Super
MCQs

 If set A has all its elements common with set B then set
A is called….set.
A. Sub
B. Overlapping
C. Disjoint
D. Super
MCQs

 If A is subset of U, then (𝐴𝑐
)𝑐
= ……
A. 𝐴𝑐
B. ∅𝑐
C. 𝑈𝑐
D. 𝐴
MCQs

General Mathematis with the Topic of SETs Story

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