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A set consisting of no element is called the empty set or null set or void set.

**Types of Sets**

There is a very special set of great interest: the
empty collection ! Why should one care about the empty collection? Consider the
set of solutions to the equation *x*^{2}+1
= 0. It has no elements at all in the set of Real Numbers. Also consider all
rectangles* *with one angle greater
than 90 degrees. There is no such rectangle and hence this describes an empty
set.

So, the empty set is important, interesting and
deserves a special symbol too.

A set consisting of no element is called the *empty set *or* null set *or* void set*.

It is denoted by *Ø* or { }.

For example,

(i)* A*={*x *:*
x *is an odd integer and divisible by 2}

*A*={
} or* **Ø*

(ii) The set of all integers between 1 and 2.

A set which has only one element is called a *singleton set*.

For example,

(i) *A *= **{x : 3 < x < 5, x*** **∈ *N**}**

(ii) *B *=
The set of all even prime numbers.

A set with finite number of elements is called a *finite set*.

For example,

1.
The set of family members.

2.
The set of indoor/outdoor games you play.

3.
The set of curricular subjects you learn in school.

4.
*A *= {*x *:*
x *is a factor of 36}

A set which is not finite is called an *infinite set*.

For example,

(i) {5,10,15,...}

(ii) The set of all points on a line.

To discuss further about the types of sets, we need
to know the cardinality of sets.

When a set is finite, it is very useful to know
how many** **elements it has. The number
of elements in a set is called the Cardinal number of the set.

The cardinal number of a set *A* is denoted by *n*(*A*)

If *A* = {1,2,3,4,5,7,9,11}, find *n*(*A*).

*Solution*

*A *=
{1,2,3,4,5,7,9,11}

Since set *A*
contains 8 elements, *n*(*A*) = 8.

Two finite sets *A*
and *B* are said to be equivalent if
they contain the same number of elements. It is written as *A* ≈ *B*.

If *A* and *B* are equivalent sets, then *n*(*A*)
= *n*(*B*)

For example,

Consider *A*
= { ball, bat} and *B* = {history,
geography}.

Here *A* is
equivalent to *B* because *n*(*A*)
= *n*(*B*) = 2.

Are *P* = { *x* : –3 ≤ *x* ≤ 0, *x* ∈ Z} and *Q* = The set of
all prime factors of 210, equivalent sets?

*Solution*

*P *= {–3,
–2, –1, 0}, The prime factors of 210 are 2,3,5,and 7 and so,* Q *= {2, 3, 5, 7}* n*(*P*) =4 and* n*(*Q*)
= 4. Therefore* P *and* Q *are equivalent sets.

Two sets are said to be equal if they contain exactly the same elements, otherwise they are said to be unequal.

In other words, two sets *A* and *B* are said to be
equal, if

(i) every element of *A* is also an element of *B*

(ii) every element of *B* is also an element of *A*

For example,

Consider the sets *A* = {1, 2, 3, 4} and *B* = {4, 2, 3, 1}

Since *A*
and *B* contain exactly the same
elements, *A* and *B* are equal sets.

A set does not change, if one or more elements of
the set are repeated.

For example, if we are given

*A*={*a*,*
b*,* c*} and* B*={*a*,* a*,*
b*,* b*,* b*,* c*} then, we write* B *= {* a, b, c, *}. Since, every element of* A *is also an element of* B *and
every element of* B *is also an element
of* A*, the sets* A *and* B *are* *equal.

Are *A* = {*x* : *x* ∈ N, 4 ≤ *x* ≤ 8} and

*B *= { 4, 5, 6, 7, 8} equal sets?

*Solution*

*A *= { 4, 5,
6, 7, 8},* B *= { 4, 5, 6, 7, 8}* *

*A *and* B *are equal sets.

Let A and B be two sets. If every element of *A* is also an element of *B,* then *A* is called a subset of *B*.
We write *A* ⊆ *B*.

A ⊆
B is read as “A is a subset of B”

Thus A ⊆ B, if a ⊆ A implies a ⊆ B.

If A is not a subset of B, we write A ⊈ B

Clearly, if A is a subset of B, then n(A) ≤ n(B).

Since every element of *A* is also an element of *B*,
the set *B* must have at least as many
elements as *A*, thus *n*(*A*)
≤ *n*(*B*).

The other way is also true. Suppose that *n*(*A*)
> *n*(*B*), then *A* has more
elements than *B*, and hence there is
at least one element in *A* that cannot
be in *B*, so *A* is not a subset of *B*.

For example,

(i) {1} ⊆ {1,2,3} (ii)
{2,4} ⊈{1,2,3}

Write all the subsets of *A* = {*a*,
*b*}.

*Solution*

*A*= {*a*,*b*}

Subsets of *A*
are Ø,{*a*}, {*b*} and {*a*, *b*}.

Let *A* and
*B* be two sets. If *A* is a subset of *B* and *A*≠*B,* then *A* is called a proper subset of *B*
and we write *A* ⊂ *B*.

For example,

If *A*={1,2,5}
and *B*={1,2,3,4,5} then *A* is a proper subset of *B* ie. *A* ⊂ *B*.

Insert the appropriate symbol ⊂ or ⊈ in each blank to make a true statement.

(i) {10, 20, 30} ____ {10, 20, 30,
40} (ii) {*p*, *q*,
*r*} _____ {*w*, *x*, *y*, *z*}

*Solution*

(i) {10, 20, 30} ____ {10, 20, 30, 40}

Since every
element of {10, 20, 30} is also an element of

{10, 20, 30, 40}, we get {10, 20, 30} ⊆ {10, 20, 30, 40}.

(ii) {p, q, r} _____ {w, x, y, z}

Since the
element p belongs to {p, q, r} but does not belong to

{w, x, y, z}, shows that {p, q, r} ⊈ {w, x, y, z}.

The fun begins when we realise that elements of
sets can themselves be sets !

That is not very difficult to imagine: the people
in school form a set, that consists of the set of students, the set of
teachers, and the set of other staff. The set of students then has many sets as
its elements: the set of students in class 1, the set of class 2 children, and
so on. So we can easily talk of sets of sets of sets of …. of sets of elements
!

Why bother? There is a particular set of sets that
is very interesting.

Let *A* be
any set. Form the set consisting of all subsets of *A*. Let us call it *B*. What
all sets does *B* contain? For one
thing, *A* is inside it, since *A* is a subset of *A*. The empty set is also a subset of *A*, so it is in *B*. If *x* is in *A*, then the singleton set {*x*}
is in *B*. (This means that *B* has at least as many elements as *A*; so *n*(*A*) is equal to *n*(*B*)).
For every pair of distinct elements *x*,
*y* in *A*, we have {*x*,*y*} in *B*. So yes, *B* has a lot
many sets. It is so rich that it gets a very powerful name !

The set of all subsets of a set *A* is called the power set of ‘*A*’. It is denoted by *P*(*A*).

For example,

(i)
If A={2, 3}, then find the power set of A.

The subsets of A are Ø , {2},{3},{2,3}.

The power set of A,

P(A) = {Ø ,{2},{3},{2,3}}

(ii)
If A = {Ø , {Ø}}, then the power set of A is { Ø , {Ø , {Ø}}, {Ø} , {{Ø}} }.

We already noted that n(A) ≤ n[P(A)]. But how big is P(A) ?
Think about this a bit, and see whether you come to the following conclusion:

(i) If n(A) = m, then n[P(A)] = 2^{m}

(ii) The number of proper subsets of a set A is
n[P(A)]–1 = 2^{m}–1.

Find the number of subsets and the number
of proper subsets of a set *X*={*a*, *b*,
*c*, *x*, *y*, *z*}.

*Solution*

Given X={*a,
b, c, x, y, z*}.Then, *n*(*X*) =6

The number of subsets =* n[P(X)]* = 2^{6} = 64

The number of proper subsets = *n[P(X)]*-1 = 2^{6}–1

= 64 – 1 =
63

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