Subject: Compulsory Maths

Depending on the number of elements contained by the sets, they are classified into the following types:

- Empty or Null set
- Unit or Singleton set
- Finite and Infinite set

**Empty or Null Set:**

A set which does not contain any elements is called empty or null set. It is denoted by a symbolΦ called 'Phi'. For example:

A = {prime number between 8 and 10}

Since there is no prime number between 8 and 10. So, set A =Φ**Unit or Singleton Set:**

A set containing only one element is called a unit set or singleton set. For example:

N = {whole number less than 1}

Since the whole number, less than 1 is 0 which is a single term. i.e N = {0}. So, the set N is said to be a unit or singleton set.**Finite and Infinite Set:**

A set which contains a finite number of elements is called a finite set. For example:

A = {odd number between 5 and 15}

Here, A = {7, 9, 11, 13}

Since we can express the given set in cardinal number i.e. n(A) = 4, It is said to be a finite set.

A set containing never-ending elements (i.e. infinite number of elements) is called an infinite set. For example:

P = {x : x is a set of prime numbers} i.e. P = {1, 2, 3, 5, 7, 11, 13,.....................}.

The given set P is so large that we cannot express it in cardinal form. It has an infinite number of elements. So, it is an infinite set.

(**Note:**Infinite sets cannot be expressed in roster form)

Depending on the types of elements containing by two or more than two sets, the relationship between them can be presented in the following ways:

- Equal sets
- Equivalent sets
- Overlapping sets
- Disjoint sets

**Equal Sets:**

Two sets are equal to be equal if both the sets have exactly the same elements. The equals set is denoted by the symbol '='. For examples:

A = {a, b, c, d, e}

B = {e, d, c, b, a}

Here, n(A) = 4 and n(B) = 4. Hence sets A and B are said to be equal set and expressed as A = B.**Equivalent Sets:**Two sets are said to be equivalent sets if they contain the same number of elements. It is denoted by the symbol '↔' or '∼'. For examples:

A = {1, 2, 3, 4 ,5}

B = {p, q, r, s, t}

Here, n(A) = 5 and n(B) = 5. Since they have the same cardinal number so they are said to be equivalent set and is expressed as A∼ B.**Overlapping Sets:**

Two or more than two sets are said to be overlapping sets if they consist at least one common element. For examples:

A = {2, 4, 6, 8, 10, 12}

B = {3, 6, 9, 12, 15}

In both sets A and B, 6 and 12 are common, so sets A and B are overlapping sets.**Disjoint Sets:**

Two or more than two sets are said to be disjoint if they do not have any of the elements common. For example:

A = {1, 2, 3, 5, 7, 11}

B = {4, 12, 16, 20}

In both sets, A and B none of the elements are common so set, A and B are disjoint.

- Empty set is denoted by a symbolΦ called 'Phi'.
- The equals set is denoted by the symbol '='.
- It is denoted by the symbol '↔' or '∼'.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

If U= {1,2,3,4,5,6,7............20} , A={1,3,5,7,9,11,13,15} and B={3,6,9,15,15,18} . Find A ∪ B .

Solution:

Given,

U= {1,2,3,4,5,6,7............20}

A={1,3,5,7,9,11,13,15}

B={3,6,9,15,15,18}

Now,

A ∪ B = {1,3,5,7,9,11,13,15} ∪ {3,6,9,15,15,18}

A ∪ B = {1,3,5,6,7,9,11,12,13,15,18}

If P={a,e,i,o,u } and Q={a,b,c,d,e}, find (a)P ∪ Q ( b) P ∩ Q (c)P - Q (d) Q-R .Represent them in venn diagrams.

Solution :

Here , P = { a,e,i,o,u} and Q={a,b,c,d,e}

a) Now , P ∪ Q = { a,b,c,d,e,i,o,u }

The shaded region represents the elements of P ∪ Q.

b) P ∩ Q = { a,e}

The shaded region represents the elements of P ∩ Q.

c) P - Q = { i,o,u}

The sheded region represents the elements of P - Q .

d) Q - P={b,c,d}

The shaded region represents the elements of Q -P .

How is null sets by ?

Null sets is denoted as phi or { } .

Define the cardinal number of sets ?

The number of elements contained by a set is known as its cardinal number .

What do mean by singleton set ?

A set containing only one element is called a unit set or singleton set.

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