Class 11 Maths Introduction to Set Basic Formulas and concept

In everyday life, we often speak of collections of objects of a particular kind, such as,
a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come
across collections, for example, of natural numbers, points, prime numbers, etc. More
specially, we examine the following collections:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x2 – 5x + 6 = 0, viz, 2 and 3.

We note that each of the above example is a well-defined collection of objects in
the sense that we can definitely decide whether a given particular object belongs to a
given collection or not. For example, we can say that the river Nile does not belong to
the collection of rivers of India. On the other hand, the river Ganga does belong to this
colleciton.
We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers.

A set is a well-defined collection of objects.
The following points may be noted :
(i) Objects, elements and members of a set are synonymous terms.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

If a is an element of a set A, we say that “ a belongs to A” the Greek symbol∈
(epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ∈ A. If ‘b’ is not
an element of a set A, we write b ∉  A and read “b does not belong to A”.

There are two methods of representing a set :

(i) Roster or tabular form

In roster form, all the elements of a set are listed, the elements are being separated
by commas and are enclosed within braces { }
For example, the set of all even
positive integers less than 7 is described in roster form as {2, 4, 6}. Some more
examples of representing a set in roster form are given below :

(a) The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}.
(b) The set of all vowels in the English alphabet is {a, e, i, o, u}.
(c) The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots

tell us that the list of odd numbers continue indefinitely

(ii) Set-builder form.

In set-builder form, all the elements of a set possess a single common property
which is not possessed by any element outside the set. For example, in the set
{a, e, i, o, u}, all the elements possess a common property, namely, each of them
is a vowel in the English alphabet, and no other letter possess this property. Denoting
this set by V, we write
V = {x : x is a vowel in English alphabet}

For example, the set
A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that
x is a natural number and x lies between 3 and 10.” Hence, the numbers 4, 5, 6,
7, 8 and 9 are the elements of the set A.

Examples

Example 1 Write the solution set of the equation x2 + x – 2 = 0 in roster form.

Solution The given equation can be written as

(x – 1) (x + 2) = 0, i. e., x = 1, – 2

Therefore, the solution set of the given equation can be written in roster form as {1, – 2}.

Example 2 Write the set {x : x is a positive integer and x2 < 40} in the roster form.

Solution The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form

is {1, 2, 3, 4, 5, 6}.

Example 3 Write the set A = {1, 4, 9, 16, 25, . . . }in set-builder form.

Solution We may write the set A as

A = {x : x is the square of a natural number}

Alternatively, we can write

A = {x : x = n2, where n ∈ N}

Subset

Consider the sets : X = set of all students in your school, Y = set of all students in your

class.
We note that every element of Y is also an element of X; we say that Y is a subset
of X. The fact that Y is subset of X is expressed in symbols as Y ⊂X. The symbol⊂
stands for ‘is a subset of’ or ‘is contained in’.
Definition 4 A set A is said to be a subset of a set B if every element of A is also an
element of B.
In other words, A ⊂ B if whenever a ∊A, then a ∈B. It is often convenient to
use the symbol “⇒” which means implies. Using this symbol, we can write the definiton
of subset as follows:
A ⊂ B if a ∈ A ⇒ a ∊ B

Φf is a subset of every set

Examples

(i) The set Q of rational numbers is a subset of the set R of real numbes, and
we write Q⊂R.
(ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56,
then B is a subset of A and we write B⊂A.
(iii) Let A = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then
A⊂B and B⊂A and hence A = B.
(iv) Let A = { a, e, i, o, u} and B = { a, b, c, d}. Then A is not a subset of B,

also B is not a subset of A.

Learn more about subset click here⇒

Power Set

Consider the set {1, 2}. Let us write down all the subsets of the set {1, 2}. We
know that Φ is a subset of every set . So, Φis a subset of {1, 2}. We see that {1}
and { 2 }are also subsets of {1, 2}. Also, we know that every set is a subset of
itself. So, { 1, 2 } is a subset of {1, 2}. Thus, the set { 1, 2 } has, in all, four
subsets, viz. Φ, { 1 }, { 2 } and { 1, 2 }. The set of all these subsets is called the
power set of { 1, 2 }.

Definition The collection of all subsets of a set A is called the power set of A. It is
denoted by P(A). In P(A), every element is a set.
Thus, as in above, if A = { 1, 2 }, then

P( A ) = { Φ,{ 1 }, { 2 }, { 1,2 }}

Universal Set

The basic setis called the “Universal Set”. Theuniversal set is usually denoted by U, and all its

subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational

numbers or, for that matter, the set R of real numbers. For another example, in human

population studies, the universal set consists of all the people in the world.

Venn Diagram

Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves

usually circles. The universal set is represented usually by a rectangle and its subsets by circles.

In Venn diagrams, the elements of the sets are written in their respective circles

Illustration 1 In Fig ,U = {1,2,3, ..., 10} is the

universal set of which

A = {2,4,6,8,10} is a subset.

The reader will see an extensive use of the

Venn diagrams when we discuss the union, intersection and difference of sets.

Learn more about venn diagram