1. SETS

2. INDEX
• ACKNOWLEDGMENT
• INTRODUCTION TO SETS
• WHAT ARE SETS
• REPRESENTATION OF A SET
• SUBSET AND SUPERSET
• TYPES OF SETS
• CARDINAL NUMBER OF A SET
• UNION OF SETS
• INTERSECTION OF SETS
• SUBRACTION OF SETS
• COMPLEMENT OF SETS
• VENN DIAGRAM
• COMMUTATIVE PROPERTIES
• ASSOCIATIVE PROPERTIES
• PROPERTIES OF PHIE (∅)
• DISTRIBUTIVE PROPERTIES
• DE-MORGAN'S LAW
• FORMULAS OF SETS
• CONCLUSION
• BIBILOGRAPHY

3. ACKNOWLEDGMENT
I would like to express my
special thanks of gratitude to
my teacher Mr. Vaibhav as well
as our principal Ms. Veena
Sharma who gave me the golden
opportunity to do this wonderful
project on the topic Sets, which
also helped me in doing a lot of
Research and i came to know
about so many new things I am
really thankful to them.
Secondly i would also like to
thank my parents and friends
who helped me a lot in finalizing
this project within the limited
time frame.

4. •The theory of sets was
developed by German
mathematician Georg
Cantor (1845-1918). He
first encountered sets while
working on “Problems on
Trigonometric Series” .
SETS are being used in
mathematics problem since
they were discovered.
INTRODUCTION
TO SETS

5. WHAT ARE SETS ?
A set is a well-defined collection
of distinct objects, considered as
an object. The arrangement of the
objects in the set does not matter. The
concept of a set is one of the most
fundamental in mathematics. Developed
at the end of the 19th century, set
theory is now a ubiquitous part of
mathematics, and can be used as a
foundation from which nearly all of
mathematics can be derived.

6. REPRESENTATION
OF A SET
There are two common ways of describing, or specifying
the members of, a set:
• ROSTER FORM
The Roster notation method of defining a set
consist of listing each member of the set. More
specifically, in roster notation , the set is denoted by
enclosing the list of members in curly brackets. For
example :
A = {4, 2, 1, 3}
B = {blue, white, red}
• SET BUILDER FORM
In set-builder notation, the set is specified as a subset of
a larger set, where the subset is determined by a
statement or condition involving the elements. For
example, a set F can be specified as follows:
F = { n | n is an integer, and 0 ≤ n ≤ 19 }
In this notation, the vertical bar ("|") means "such that",
and the description can be interpreted as "F is the set
of all numbers n, such that n is an integer in the
range from 0 to 19 inclusive". Sometimes
the colon (":") is used instead of the vertical bar.

7. SUBSET
AND
SUPERSET
If every element of set A is also in B, then A is said to be
a subset of B, written A ⊆ B (pronounced A is contained in
B). Equivalently, one can write B ⊇ A, read as B is a superset of A, B
includes A, or B contains A. The relationship between sets established
by ⊆ is called inclusion or containment. Two sets are equal if they
contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.
If A is a subset of B, but not equal to B, then A is called a proper
subset of B, written A ⊊ B, or simply A ⊂ B (A is a proper
subset of B), or B ⊋ A (B is a proper superset of A, B ⊃ A). For
Examples:
The set of all humans is a proper subset of the set of all
mammals, {1, 3} ⊆ {1, 2, 3, 4}, {1, 2, 3, 4} ⊆ {1, 2, 3, 4}, etc.

8. TYPES OF SETS
Universal set : Any set which is a superset of all the sets under consideration is
said to be universal set and is either denoted by omega or S or U. For example
: Let A = {1, 2, 3}, C = { 0, 1} then we can take S = {0, 1, 2, 3, 4, 5, 6, 7, 8,
9} as universal set.
Finite Set : A set consisting of a natural number of objects, i.e. in which number
element is finite is said to be a finite set. Consider the sets. A = { 5, 7, 9, 11}
and B = { 4 , 8 , 16, 32, 64, 128}. Obviously, A, B contain a finite number of
elements, i.e. 4 objects in A and 6 in B. Thus they are finite sets.
Infinite set : f the number of elements in a set is finite, the set is said to be an
infinite set. Thus the set of all-natural number is given by N = { 1, 2, 3, ...} is an
infinite set. Similarly the set of all rational number between 0 and 1 given by
: A = { x : x ∈ Q, 0 < x <1 } is an infinite set.

9. Equal set : Two sets A and B consisting of the same elements are said to be
equal sets. In other words, if an element of the set A sets the set A and B are
called equal i.e. A = B.
Null set/ empty set : A null set or an empty set is a valid set with
no member. A= {}/ ∅ cardinality of A is 0. There are two
popular representation either empty curly braces { } or a special symbol phie
(∅). This A is a set which has null set inside it.
Singleton Set : A singleton set is a set containing exactly one element. Example:
Let B = {x : x is an even prime number} Here B is a singleton set because there
is only one prime number which is even, i.e., 2. A = {x : x is neither prime
nor composite} It is a singleton set containing one element, i.e., {1}.

10. Power set : The power set of a set S is the set of all subsets of S. The power set contains S itself
and the empty set because these are both subsets of S. 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 is usually written
as P(S). The power set of a finite set with n elements has 2n elements. For example, the set {1, 2,
3} contains three elements, and the power set shown above contains 2 X 2 X 2 = 8 elements
Equivalent Sets : An equivalent set is simply a set with an equal number of elements. The sets do
not have to have the same exact elements, just the same number of elements. For Example : Set
A: {A, B, C, D, E}, Set B: {January, February, March, April, May}. Even though Sets A and B have
completely different elements (Set A comprises of letters, and Set B comprises of months of the
year), they have the same amount of elements, which is five. Set A contains five letters and Set B
contains five months, that makes them equivalent sets.
Disjoint Sets : Two sets are said to be disjoint sets if they have no element in common.
Equivalently, two disjoint sets are the sets whose intersection is the empty set. For example, {1, 2,
3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of
more than two sets is called disjoint if any two distinct sets of the collection are disjoint.

11. CARDINAL NUMBER OF A SET
• The number of distinct elements in a finite
set is called its cardinal number. It is
denoted as n(A) and read as ‘the number
of elements of the set’.
For example:
(i) Set A = {2, 4, 5, 9, 15} has 5 elements.
Therefore, the cardinal number of set A
= 5. So, it is denoted as n(A) = 5.
(ii) Set B = {w, x, y, z} has 4 elements.
Therefore, the cardinal number of set B
= 4. So, it is denoted as n(B) = 4.
(iii) Set C = {Florida, New York, California}
has 3 elements.
Therefore, the cardinal number of set C = 3.
So, it is denoted as n(C) = 3.

12. UNION OF SETS
• In set theory, the union (denoted by ∪) of a
collection of sets is the set of all elements in the
collection. It is one of the fundamental operations
through which sets can be combined and related to
each other.
• The union of two sets A and B is the set of
elements which are in A, in B, or in both A and B.
In symbols,
A U B = { x : x ∈ A or x ∈ B }.
• For example, if A = {1, 3, 5, 7} and B = {1, 2,
4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A
more elaborate example (involving two infinite
sets) is:
A = {x is an even integer larger than 1}
B = {x is an odd integer larger than 1}
A U B = {2,3,4,5,6, . . . }
• Sets cannot have duplicate elements, so the union
of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}.
Multiple occurrences of identical elements have no
effect on the cardinality of a set or its contents.

13. INTERSECTION OF SETS
• The intersection of two sets A and B, denoted by A ∩ B, is
the set containing all elements of A that also belong to B (or
equivalently, all elements of B that also belong to A). The
intersection of two sets A and B, denoted by A ∩ B, is the set of all objects
that are members of both the sets A and B. In symbols, A ∩ B={ x :
x ∈ A and x ∈ B .} .That is, x is an element of the
intersection A ∩ B if and only if x is both an element of A and
an element of B. Intersection is written using the sign "∩"
between the terms; that is, in infix notation. For example,
•{1,2,3} ∩ {2,3,4} = {2,3}
•{1,2,3} ∩ {4,5,6} = ∅
• Z ∩ N = N
• {x ∈ R : x is an even prime number } ∩ N = 2

14. SUBRACTION OF SETS
Subraction between sets is a way
of modifying a set by removing
the elements belonging to another
set. Subtraction of sets is indicated
by either of the symbols – or . For
example, A minus B can be written
either A – B or A B.
For example : A = { 1,2,3,4,5,6} and
B = { 2,3,5,7,11,13,17}
So, A – B = {1,2,3,4,5,6} -
{2,3,5,7,11,13,17} = {1,4,6}

15. COMPLEMENT OF A SET
• If A is a set, then the complement of
A is the set of elements not in A,
within a larger set that is implicitly
defined. In other words, let U be a set
that contains all the elements under
study; if it is no need to mention U,
either because it has been previously
specified, or it is obvious and unique,
then the absolute complement of A is
the relative complement of A in U:
A' = U – A .
Formally:
A' = { x ∈ U , x ∉ A }
The absolute complement of A is
usually denoted by A'

16. VENN DIAGRAM
A Venn diagram or set diagram is a
diagram that shows all possible
logical relations between a finite
collection of sets. Venn diagrams
were conceived around 1880 by
John Venn. They are used to teach
elementary set theory, as well as
illustrate simple set relationships in
probability, logic, statistics
linguistics and computer
science. Venn consist of rectangles
and closed curves usually circles. The
universal is represented usually by
rectangles and its subsets by circle.

17. COMMUTATIVE PROPERTIES
• The Commutative Property for Union and the Commutative Property
for Intersection say that the order of the sets in which we do the
operation does not change the result.
• General
Properties: A ∪ B = B ∪ A and A ∩ B = B ∩ A.
• Example: Let A = {x : x is a whole number between 4 and 8} and B =
{x : x is an even natural number less than 10}.
Then A ∪ B = {5, 6, 7} ∪ {2, 4, 6, 8} = {2, 4, 5, 6, 7, 8} = {2, 4, 6,
8} ∪ {5, 6, 7} = B ∪ A
and A ∩ B = {5, 6, 7} ∩ {2, 4, 6, 8} = {6} = {2, 4, 6, 8} ∩ {5, 6, 7}
= B ∩ A.

18. ASSOCIATIVE PROPERTIES
The Associative Property for
Union and the Associative Property
for Intersection says that how the sets
are grouped does not change the
result.
General
Property: (A ∪ B) ∪ C = A ∪ (B ∪ C)
and (A ∩ B) ∩ C = A ∩ (B ∩ C)
Example: Let A = {a, n, t}, B = {t, a,
p}, and C = {s, a, p}.
Then (A ∪ B) ∪ C = {p, a, n, t} ∪ {s, a,
p} = {p, a, n, t, s} = {a, n, t} ∪ {t, a,
p, s} = A ∪ (B ∪ C)
and (A ∩ B) ∩ C = {a, t} ∩ {s, a, p} =
{a} = {a, n, t} ∩ {a, p} = A ∩ (B ∩ C).

19. PROPERTIES OF PHIE (∅)
The Identity Property for
Union says that the union of a
set and the empty set is the set,
i.e., union of a set with the
empty set includes all the
members of the set.
General
Property: A ∪ ∅ = ∅ ∪ A = A
Example: Let A = {3, 7, 11}
and B = {x : x is a natural
number less than 0}.
Then A ∪ B = {3,
7, 11} ∪ { } = {3, 7, 11}.
The Intersection Property of the
Empty Set says that any set
intersected with the empty set
gives the empty set.
General
Property: A ∩ ∅ = ∅ ∩ A = ∅.
Example: Let A = {3, 7, 11}
and B = {x : x is a natural
number less than 0}.
Then A ∩ B = {3,
7, 11} ∩ { } = { }.

20. DISTRIBUTIVE PROPERTIES
The Distributive Property of Union over Intersection and the Distributive Property of
Intersection over Union show two ways of finding results for certain problems mixing
the set operations of union and intersection.
General Property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) =
(A ∩ B) ∪ (A ∩ C)
Example: Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}. Then, A ∪ (B ∩ C) = {a, n,
t} ∪ {a, p} = {p, a, n, t} = {p, a, n, t} ∩ {p, a, n, t, s} = (A ∪ B) ∩ (A ∪ C)
and A ∩ (B ∪ C) = {a, n, t} ∩ {t, a, p, s} = {a, t} = {a, t} ∪ {a} = (A ∩ B) ∪ (A ∩ C)

21. DE - MORGAN'S LAW
De Morgan's laws is a pair of transformation
rules that are both valid rules of inference.
They are named after Augustus De Morgan, a
19th-century British mathematician. The rules
allow the expression
of conjunctions and disjunctions purely in
terms of each other via negation. This law
states that : 1.
(AUB)'=A'∩B' 2. (A∩B)'=A'UB'
For example : Let U = {1,2,3, 4,5,6,7,8}, P =
{4,5,6} and Q = {5,6,8}. P ∪ Q = {4,5,6} ∪
{5,6,8} = {4,5,6,8}. So, (P∪Q)' = {1,2,3,7} …. (i)
Now P = {4,5,6}; so, P' = {1,2,3, 7,8}
and Q = {5, 6, 8}; so, Q' = {1, 2, 3, 4, 7}
P' ∩ Q' = {1, 2, 3, 7, 8} ∩ {1, 2, 3, 4, 7}
Therefore, P' ∩ Q' = {1, 2, 3, 7} ….. (ii)
Combining (i)and (ii) we get; (P∪Q)'=P'∩Q'.

22. FORMULAS
OF SETS
1. U U A = U
2. U ∩ A = A
3. A U A = A
4. A ∩ A = A
5. A U A' = U
6. A ∩ A' = ∅
7. (A')' = A
8. ∅' = U
9. U' = ∅
10. n(A U B) = n(a) + n(B) - n(A ∩ B)
11. A – B = A ∩ B'
12. n(A – B) = n(A) - n(A ∩ B)
13. n(A) + n(A') = n(U)
14. n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n( B ∩ C) - n(A ∩ C) + n( A ∩ B ∩ C)

23. CONCLUSION
Set theory is used in almost every discipline including engineering,
business, medical and related health sciences, along with the natural
sciences.
In business operations, it can be applied at every level where intersecting
and non-intersecting sets are identified.
For example, the sets for warehouse operations and sales operations are
both intersected by the inventory set. To improve the cost of goods sold,
the solution might be found by examining where inventory intersects both
sales and warehouse operations.
For Security purpose, risk management and system analysis.

24. BIBILOGRAPHY
www.google.com
www.googleimages.com
En.wikipedia.org
www.slideshare.net
www.math-only-math.com
www.mathwords.com
www.study.com
www.includehelp.com

25. NAME :
VIHAAN
BHAMBHANI
CLASS :
XI