In mathematics, a set is defined as a collection of distinct, well-defined objects forming a group. There can be any number of items, be it a collection of whole numbers, months of a year, types of birds, and so on. Each item in the set is known as an element of the set. We use curly brackets while writing a set.

1. SETS
CLASS-
XI(MATHEMATICS)

2. TOPICS:
 Introduction of sets
 Representation of sets
 Types of sets
 Sets in mathematical world
 Subsets and Proper subsets
 Set operations
 Complement of set
 Law in set theory
 Problems based on two and three sets

3. INTRODUCTION: SETS
 Father of Set theory- George Cantor(1845-1918).
 A well-defined collection of distinct objects is called a set.
 Sets is to be represented by capital letters only i.e. A, B, C, D, E----Z.
 The elements of a set are represented by small letters i.e. a, b, c, d, e-----z.
 Greek symbol epsilon(∈) is used to denote the phrase ‘belongs to’.
 a is member of set A =a ∈ A.
 b is not member of set A= b∉ A
 A set may or may not contain elements. A set having no elements is written as {
}.

4. REPRESENTATION OF SETS
ROSTER FORM
SET BUILDER FORM

5. ROSTER FORM
ROSTER FORM- A set is represented by
listing all its elements in curly brackets
and separating them by commas.
Note: Sequence doesn’t matter in roster
form.
Letters/objects is not to be repeated.
1. E.g. V= {a, e, o, i, u}
2. E.g. Word-MATHEMATICS- X={M, A, T,
H, E, I, C, S}

6. SET BUILDER FORM
• A set is represented by stating all the properties which is
satisfied by the elements of the set and not by any other
element outside the set.
• Note: Mostly used when no of elements are many.
• 1.E.g.- V={ x: x is a vowel in English alphabet}
• 2.Eg.- A={ x: x is an integer and -3<x>7}.
• 3.E.g.- E={ x: x is a multiple of 3 such that x<15}.

7. TYPES OF SETS
FINITE SET
SINGLETON
SET
NULL
SET
EQUAL SET
INFINITE
SET
EQUIVALENT
SET

8. SETS IN MATHEMATICAL WORLD
• N: set of all natural numbers={1, 2, 3, 4--------}.
• W: set of all whole numbers={0,1,2,3,4--------}
• Z: set of all integers{-∞,−−−− −, −1,0,1,2 −−− −∞}
• Q: set of all rational numbers(
𝑃
𝑄
𝑓𝑜𝑟𝑚, 𝑃 𝑎𝑛𝑑 𝑄 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑄 ≠ 0)
• T: set of all irrational numbers
(not in
𝑃
𝑄
𝑓𝑜𝑟𝑚, 𝑃 𝑎𝑛𝑑 𝑄 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑄 ≠ 0)
• R: set of all real numbers(rational +irrational numbers)
• 𝑍+: set of all positive integers
• 𝑄+: set of all positive rational numbers
• 𝑅+
: set of all positive real numbers
• 𝑅0: set of all non- zero real numbers

9. FINITE SET and INFINITE SET
 FINITE SET- It contains only finite number of elements.
 E.g. The set {2, 4, 5, 10} is a finite set only 4 elements.
 INFINITE SET- If it doesn’t contain finite number of elements.
 E.g. The set of all even numbers.
 The sets N, W, Z, Q, T, R are all infinite sets.
 CARDINALITY OF SET- If A is a finite set then number of elements in A is
called the cardinality of the finite set A. It is denoted by n(A).

10. NULL SET- If a set does not contain any element is
called null set or void set or empty set. It is denoted
by ∅.
The set{0} is not a null set as it contains one element
i.e. 0.
SINGLETON SET-
If a set contains only one element is called singleton set.
NULL SET and SINGLETON SET

11. EQUIVALENT SETS and EQUAL SETS
 Equivalent sets- Two sets A and B, if the
elements of A can be paired with the
elements of B so that each element of
A there corresponds exactly one
element of B and to each element of B
there corresponds exactly one element
of A.
 Note- Finite sets of A and B are
equivalent iff the number of elements
in A and B are equal.

12. EQUAL SETS
• Equal sets- Two sets A and B are said to be
equal if they have exactly the same elements
and vice-versa i.e. A=B.
• If x ∈A ⇒x ∈ B and x ∈ B ⇒ x ∈ A.
If sets A and B are equal then A=B.
• Note- 1. For two sets to be equal, the order of
elements need not be same.
2. Equal sets are equivalent sets but equivalent
sets may not be equal sets.
e.g. {2,4,5} and {3,7,8} are equivalent sets but
not equal sets.

13. SUBSETS
A set A is said to be a subset of set B ,if every
element of A is an element of B.
If a is a subset of b, a⊆b
∴ A ⊆ B if x∈ A⇒ x ∈ B
If A ⊆ B, then B is a superset of A and B⊇ A
If A is not a subset of B, ∃ atleast one element in
A, which is not in B, A ⊈B

14. PROPERTIES OF SUBSETS
 The null set is subset of every set.
Let A be any set.
∅ ⊆ 𝐴, because there is no element in ∅ which is not in A.
 Every set is subset of itself.
Let A be any set.
x ∈ A ⇒ x ∈ A though trivially ∴ 𝐴 ⊆ 𝐴.
 If A ⊆ B and B ⊆ C, then A ⊆ C.
Let x ∈ A
∴ x ∈ B (A ⊆ B)
∴ x ∈ C (B ⊆ C)
∴ A ⊆ C

16. PROPER
SUBSETS
■ A set A is said to be a proper subset of
set B, if A is a subset of B and A is not
equal to B.
■ If A is a proper subset of B, then A ⊂ B.
■ If A is a proper subset of B, then every
element of A is in B and B must have
atleast one element which is not in A.

17. SUBSETS OF THE SET R OF REAL NUMBERS
 N: set of all natural numbers={1, 2, 3, 4--------}.
 W: set of all whole numbers={0,1,2,3,4--------}
 Z: set of all integers{-∞,−−−− −, −1,0,1,2 −−− −∞}
 Q: set of all rational numbers(
𝑃
𝑄
𝑓𝑜𝑟𝑚, 𝑃 𝑎𝑛𝑑 𝑄 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑄 ≠ 0)
 T: set of all irrational numbers
(not in
𝑃
𝑄
𝑓𝑜𝑟𝑚, 𝑃 𝑎𝑛𝑑 𝑄 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑄 ≠ 0),{x : x∈ 𝑅 𝑏𝑢𝑡 𝑥 ∉ Q}

18. z
INTERVALS
 Open Interval- If a and b are real numbers such that a<b, then set of all real numbers between a and
b is called open interval from a to b.
 Written in form- (a, b)
 The points a and b are called end points of the open interval(a, b).
 An open interval does not contain its end points, i.e. a ∉ ( a, b) and b ∉( a, b).

19. z
 Closed Interval- If a and b are real numbers such that a< b, then set of all real numbers
between a and b is called closed interval from a to b.
 Written in form- [a, b]
 The points a and b are called end points of the closed interval[ a, b].
 An open interval contain its end points, i.e. a ∈ [ a, b] and b ∈[ a, b].

25. WORKING RULES
Rule I- The set A is empty if it contains no element.
Rule II- The set A is a singleton set if it contains exactly one element.
Rule III- The sets A and B are equal if every element of A is in B and every
element of B is in A.
Rule IV- The set A is a subset of set B if every element of A is in B.
Rule V- The set A is a proper subset of set B if every element of A is in B
and B has atleast one element which is not in A.

26. SET OPERATIONS – VENN DIAGRAMS
 English logician-John
Venn(1834-1923).
 Relationships between sets
can be easily visualised by
means of diagrams called
Venn Diagrams.

27. UNIVERSAL SET
■ A set X is called a Universal set if every
set under consideration is a subset of X.
■ It is not a fixed set.
■ It varies from situation to situation.
■ It is denoted by ‘U’.
■ A Universal set is depicted by the
interior of rectangle whereas subsets of
universal set are depicted by interior of
circles, ellipses etc.

28. UNION OF SETS
• The union of two sets A and B is defined as the
set of all those elements which are in either A
or B or both.
• The union of sets A and B is denoted as A ∪ B.
• Symbol ‘∪’ is used to denote the ‘Union’.
• A ∪ B = {x: x ∈ 𝐴 or x ∈ B}
• If x ∉ A ∪ B, x ∉ A and x ∉ B.
• The union of n sets 𝐴1, 𝐴2, 𝐴3,-------- 𝐴𝑛 is
denoted by 𝐴1 ∪ 𝐴2 ∪ 𝐴3 ∪-------- ∪ 𝐴𝑛 or
𝑖=1
𝑛
𝑈 𝐴𝑖.

29. The intersection of two sets A and B is defined as the
set of all those elements which are in both A and B.
The intersection of sets a and b is denoted as A ∩ B.
Symbol ‘∩’ is used to denote the ‘intersection’.
A ∩ B = {x: x ∈ 𝐴 or x ∈ B}
If x ∉ A ∩ B, x ∉ A and x ∉ B.
The union of n sets 𝐴1, 𝐴2, 𝐴3,-------- 𝐴𝑛 is denoted by
𝐴1 ∩ 𝐴2 ∩ 𝐴3 ∩--------∩ 𝐴𝑛 or 𝑖=1
𝑛
∩ 𝐴𝑖.
INTERSECTION OF SET

30. DISJOINT SETS
 Two sets A and B are said to be
disjoint sets if there is no element
which belongs to both A and B.
 If A and B are disjoint sets, then A
∩ B =𝜙

32. DIFFERENCE OF SETS
 The difference of two sets A and B in
this order is the set of all those
elements of A which are not in B.
 The difference of A and B is denoted
by A− B.
 A− B or B− A are not equal sets.
These are always disjoint sets.

33. WORKING RULES