There are two methods to represent a set

(i) Roster Method In this method, elements are listed and put within a the braces {} and separated by commas. This method is also known as tabular method or listing method.

(ii) Set Builder Method In this method, we list the property or properties satisfied by the elements of set and write it as

`A= {x: P(x)}` or {xl P(x)}

It is read as 'A is the set of all members x such that x has the property P(x)'. The symbol ':' or '|' stands for 'such that'.

This method is also known as rule method or property method.

e. g. A= {1, 2, 3, 4, 5} = {`x : in N` and ` x<= 5`}

Finite and Infinite Sets Sets containing finite number of elements are called finite sets. e.g. A ={a, e, i. o, u)

A set containing infinite number of elements is called infinite set. e.g. A= Set of points on a line.

Order of a Finite Set The number of elements in a finite set is called the order of the set A and is denoted by n (A). It is also called cardinal number of the set.

e.g. A= {2, 4, 6, 8}

`n(A) = 4`

Null Set or Empty Set A set which contains no element, is called an empty set or null set. It is also called a void set. The null set is denoted by the symbols `phi`.

`:. phi = { x: x is an integer between 2 and 3)

Singleton Set A set consisting of a single element is called a singleton set.

A= {a} is a singleton.

Subset If each clement of a set A is also an element of a set B. A is called a subset of B. We write it as A

Definition : A set which does not contain any element is called the empty set or the null set or the void set, is denoted by the symbol `φ` or { }.

E.g.

(i) Let `A =` {`x : 1 < x < 2, x` is a natural number}. Here is no natural number between 1 and 2. So `A` is the empty set.

(ii) `B =` {`x : x^2 – 2 = 0` and `x` is rational number}. Then `B` is the empty set because the equation `x^2 – 2 = 0` is not satisfied by any rational value of `x`.

Definition : A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Consider some examples :

(i) Let W be the set of the days of the week. Then W is finite.
(ii) Let S be the set of solutions of the equation `x^2 –16 = 0`. Then `S` is finite.

(iii) Let G be the set of points on a line. Then G is infinite.


Note : All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

`"Cardinality"` of a finite Set is defined as total number of elements in a set.

Definition : Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write `A ≠ B`.

We consider the following examples :

(i) Let `A = {1, 2, 3, 4}` and `B = {3, 1, 4, 2}`. Then `A = B`.

(ii) Let `A` be the set of prime numbers less than 6 and P the set of prime factors of `30`. Then A and P are equal, since 2, 3 and 5 are the only prime factors of `30` and also these are less than `6`.

Note : A set does not change if one or more elements of the set are repeated. For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of A is in B and vice-versa .

- Definition : A set A is said to be a subset of a set B if every element of A is also an element of B.

- We write as, `A subseteq B,` and is called A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A.

E.g. `A subseteq B` if `a ∈ A ⇒ a ∈ B`

- We read the above statement as “A is a subset of B if `a` is an element of `A` implies that `a` is also an element of `B`”. If A is not a subset of B, we write `A ⊈ B`.

- If Every element of B is also in A and if every element of A may or may not be in B. In this case, A and B are the same sets so that we have `A subseteq B` and `B subseteq A ⇔ A = B`, where `“⇔”` is a symbol for two way implications, and is usually read as if and only if .

Note : It follows from the above definition that every set A is a subset of itself, i.e., `A subseteq A`. Since the empty set `φ` has no elements, we agree to say that `φ` is a subset of every set.

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.
E.g. If `A = { 1, 2 }`, then `P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}` Also, note that `n [ P (A) ] = 4 = 2^2`

In general, if `A` is a set with `n(A) = m`, then it can be shown that `n [ P(A)] = 2^m`

A set which is such that all the sets under consideration an: its subsets, is called the uni versa! set or universe and is denoted by U.

e.g. If A= {2, 4, 6, 8}, B= {1, 3, 5, 7}

Then, U = (I, 2, ;), 4, 5, 6, 7, 8}

Complement of a Set with Respect to the Universal Set Complement of a set A is the set which contains all those elements of the universal set which are not in A. It is denoted by A' or A.

e.g. U = {1, 2, 3, 4, 5, 6, 7} and A= {2, 4, 5}

A'= {1,3,G, 7}

- A Venn diagram is a diagram that uses circles to illustrate the relationships among sets.
- These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.


E.g. 1 : In Fiig. `U = {1,2,3, ..., 10}` is the universal set of which `A = {2,4,6,8,10}` is a subset.




E.g. 2 In Fig , `U = {1,2,3, ..., 10}` is the universal set of which

`A = {2,4,6,8,10}` and `B = {4, 6}` are subsets, and also `B ⊂ A.`

Union of sets :

- The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). The common elements being taken only once.
- Symbolically, we write `A ∪ B` and usually read as ‘A union B’.

`A ∪ B = { x : x ∈ A` or `x ∈B }`

Eg., Let `A = { 2, 4, 6, 8}` and `B = { 6, 8, 10, 12}`. Find `A ∪ B`.

`A ∪ B = { 2, 4, 6, 8, 10, 12}`

Some Properties of the Operation of Union :

(i) `A ∪ B = B ∪ A` (Commutative law)

(ii) `( A ∪ B ) ∪ C = A ∪ ( B ∪ C)` (Associative law )

(iii) ` A ∪ φ = A` (Law of identity element, `φ` is the identity of `∪`)

(iv) `A ∪ A = A` (Idempotent law)

(v) `U ∪ A = U` (Law of `U`)

- The intersection of two sets A and B is the set of all those elements which belong to both A and B.The symbol ‘∩’ is used to denote the intersection.
- Symbolically, we write `A ∩ B =` {`x : x ∈ A` and `x ∈ B`}

- The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write `A ∩ B =` {`x : x ∈ A` and `x ∈ B`}.

`A = { 2, 4, 6, 8}` and `B = { 6, 8, 10, 12}`. A ∩ B = { 6, 8 }.

(i) `A ∩ B = B ∩ A` (Commutative law).

(ii) `( A ∩ B ) ∩ C = A ∩ ( B ∩ C )` (Associative law).

(iii) ` φ ∩ A = φ, U ∩ A = A` (Law of `φ` and `U`).

(iv) `A ∩ A = A` (Idempotent law)

(v) `A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )` (Distributive law ) i. e., `∩` distributes over `∪`(Proof is shown in Fig. )

(v) `A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C )`

This can be seen easily from the following Venn diagrams

- The difference of the sets A and B in this order is the set of elements which belong to A but not to B.
- Symbolically, we write A – B and read as “ A minus B”

Eg. Let `A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }`.

`A – B = { 1, 3, 5 }`, since the elements `1, 3, 5` belong to A but not to `B`
and `B – A = { 8 }`, since the element 8 belongs to B and not to A. We note that `A – B ≠ B – A`.

Remark : The sets `A – B, A ∩ B` and `B – A` are mutually disjoint sets, i.e., the intersection of any of
these two sets is the null set as shown in Fig

`"Also note : A-B also represented as A/B"`

- Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.
- Symbolically, we write A′ to denote the complement of A with respect to U.

Thus, A′ = {x : x ∈ U and x ∉ A }. Obviously A′ = U – A


E.g. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}.

A′ = { 2, 4, 6, 8,10 }.


Some Properties of Complement Sets :

(i). `A ∪ A′ = U`
(ii) `A ∩ A′ = φ` (Complement laws)
(iii) `(A ∪ B)´ = A′ ∩ B′`
(iv) `(A ∩ B )′ = A′ ∪ B′` (De Morgan’s law)
(v) `(A′ )′ = A` (Law of double complementation )
(vi). `φ′ = U` and `U′ = φ` (Laws of empty set and universal set)

1. `A-B = A∩B'`

Proof of this property is shown in Figure with the help of venn diagram



Similarly

2. `A-(B∩C) = (A-B) ∪ (A-C)`
3. `A-(B∪C) = (A-B) ∩ (A-C)`
4.`A∩(B-C) = (A∩B) - (A∩C)`