2. INTRODUCTION...
Today the concept of sets is being used in almost
every branch of mathematics. Sets are used to
define the concepts of relations and functions.
The study of geometry, sequences, probability,
etc. requires the knowledge of sets. The theory of
sets was developed by German mathematician
(1845-1918). He first encountered
sets while working on “problems on
trigonometric series ’’.
3. SETS
In everyday life, we see various kinds of collections of objects. In
mathematics also, we come across various types of collections
such as, the vowels in English alphabet, types of triangles, the
states in India.
Given below are some examples of sets used in mathematics:
Thus, a set is a well -defined collection of objects.
It’s notation is { }.
4. Important points about sets…
I. Sets are usually denoted by capital letters X,D,M,R,Y,A, etc.
II. The elements of a set are represented by a small letters
a,d,g,y,o, etc.
III. If a is an element of A we denote it by a ∈ A. If b is not an
element of A then it’s denoted by b ∉ B.
IV. A set which does not contain any element is called an
V. A set which contains a definite number of elements is called
and sets which contains infinite elements are called
VI. A set which contains only one element is known as
VII. Two sets X and Y are said to be if both have same
elements.
5. Methods of representing sets:
All the elements of the set are listed,
elements are separated by commas and enclosed within
brackets{ } e.g. the set of the multiples of 2 less than 10
in roster form is as {2,4,6,8}.
All the elements of a set possess a
single common property which is not possessed by any
element outside the set e.g. in the set {a,e,i,o,u} all the
elements possess a common property i.e. each of them is
a vowel in the English alphabet and no other letter
possesses this property. Denoting the set v,
V= {x : x is a vowel in the English alphabet}
6. Operations On Sets:
• Let A and B be two sets. The union of A and B is
the set of those elements which belong to A or B
or both.
• AB={x:xϵB}
• x ϵ AB ↔ x ϵ A or x ϵ B
7. • Let A and B be two sets. The intersection of A
and B is the set of all those elements that belong
to both A and B .
• A B={x:x ϵ A and x ϵ B}
• x ϵ A B ↔ x ϵ A and x B
8. • Two sets A and B are said to be disjoint if they
do not have any common element. i.e AB=ф .
• If AB ≠ ф, then A and B are said to be
intersecting or overlapping sets.
9. • Let A and B be two sets. The difference of A and
B, written as A-B, is the set of all elements of A
which do not belong to B.
⇒ A-B={x:x ϵ A and x ∉ B}
⇔ x ϵ A-B ↔ x ϵ A and x ∉ B
10. • Let A and B be two sets. The symmetric
difference of sets A and B is the set (A-B)(B-A).
• It is denoted by AΔB.
• AΔB=(A-B)(B-A) ={x:x ∉ AB}.
11. • Let U be the universal set and let A be a set such
that A⊂U. Then the set of elements of U which
are not in A is known as the complement of A.
Denoted by A’.
• A’={x ϵ U : x ∉ A}
• xϵ A’ ↔ x ∉ A.
12. Subsets
•Let A and B be two sets. If every element of A is
present in B also, then A is known as the subset of
B. A B.
And B is known as the superset of A. A⊇B
•A B if and only if every element of A is also
an element of B.
•A B x (xA xB)
13. Subsets
Important Points:
No. of subsets of A= 2ⁿ.
n= no. of elements in A
ф is the subset of every set.
Every set is a subset of itself.
Every set is a superset of itself
15. Power Set:
• If A is any set, then power set P(A) is the set of
all subsets of A.
• Examples:
1. A = {x, y, z}
• P(A) = {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z},
{x, y, z}}
2.A =
• P(A) = {}
• Note :n(A) = 0, n(P(A)) = 1
16. Laws of Algebra of Sets
• If A is any set, then:
i) A∪A’ =U ii) A∩A’=ф
iii) A∪ф= A iv) (A’)’=A
• If A and B are two sets, then: (De-Morgan’s Law)
i) (A∪B)’= A’∩B’ ii) (A∩B)’=A’∪B’
• If A,B, and C are three sets, then:
i) ( A∪B)∪C= A∪(B∪C) ii)A∩(B∩C)=(A∩B)∩C
iii) A∪(B∪C)=(A∪B)∩(A∪C)
iv) A∩(B∪C)=(A∩B)∪(A∩C)
