Different Type of Sets

The different types of sets are described below with examples.

Finite Set:

A set is called a finite set if the members of the set can be counted.

Examples: (i) , which has 4 members.

(ii) , which has 10 members.

Infinite Set:

A set is called an infinite set if it it has countless members.

Examples: (i) The set of whole numbers.

(ii)

It is not easy to write infinite sets in the tabular form because it is not possible to make a list of an infinite number of members. The example (i) can be written in the tabular form as

The example (ii) can be written in the tabular form as

.

Empty set:

A set which has no members is called an empty set or a null set. The empty set is denoted by or .

Example: The set is an empty set because

.

Note: An empty set is also a finite set.

Singleton Set:

A set which contains only one member is called a singleton set.

Examples: (i) {  is neither prime nor composite }. This is a singleton set containing one element, i.e. 1.

(ii) { is an even prime number} $. This is a singleton set because there is only one even number which is prime, i.e., 2.

Pair Set:

A set which contains only two members is called a pair set.

Example: . This is a pair set because there are only two members, i.e, 0 and 1.

Universal Set:

The set of all objects under consideration is the universal set for that discussion. For example, if A, B, C, etc. are the sets in our discussion then a set which has all the members of A, B, C, etc., can act as the universal set. Clearly, the universal set varies from problem to problem. It is denoted by U or .

Example: If the sets involved in a discussion are sets of some natural numbers then the set of all natural numbers may be regarded as the universal set.

Cardinal Number of a Set:

The cardinal number of a finite set A is the number of distinct members of the set and it is denoted by . The cardinal number of the empty set is 0 because has 0 members. So, . And the cardinal number of an infinite set cannot be found because such a set has countless members.

Examples: (i) If then .

(ii) If {  is a letter of the word PATNA} then because A in the tabular form is .

Note: If , we call set A a singleton set.

If , we call set A an pair set.

Equivalent Sets:

Two finite sets with an equal number of members are called equivalent sets. If the sets A and B are equivalent, we write and read this as “A is equivalent to B”.

if .

Examples: Let , and  { is a letter of the word DOOR} .

Then, and because . So .

Subsets:

If two sets A and B are such that every member of A is also a member of B then we say that A is a subset of B. This is denoted by . the fact that the set A is a subset of B can also be expressed by saying B is a superset of A. We denote this by .

Example: Let and

Then, and . But and . So, .

Similarly, , and .

Now, and . Also, and . Thus, all the members of A are members of C. So, . Also, .

Note: (i) Since the empty set does not have any member, it is a subset of every other set.

(ii) By the definition of a subset, every set A is its own subset, i.e., .

Equal Sets:

Two sets A and b are equal if every member of A is a member of B, and every member of B is a member of A. In other words, two sets A and B are equal if and . This is denoted as

Example: Let and

Writing in the tabular form,

Here, every member of A is a member of B, i.e.,

Also, every member of B is a member of A, i.e.,

So, . The sets A and B are equal sets.

Exercise:

1. Write the following sets in tabular form and find their cardinal numbers:

(i)

(ii) { is a prime number of digit one}

(iii)  { is a two-digit number divisible by 15}

(iv)  { is a letter of the word BHARATI}

2. Write the following sets in set-builder form:

(i)

(ii)

(iii)

(iv)

3. Identify the finite and infinite sets. Find the cardinal number of the finite sets.

(i)

(ii) 

(iii)

(iv)

(v) 

(vi)

4. Identify the empty set, singleton set and pair set:

(i)

(ii)

(iii)

(iv)

5. Let  { is a letter of the word MINISTER}

And  { is a letter of the word SINISTER}

State which of the following are true and which are false:

(i)

(ii)

(iii)

(iv)

6. Let . State which of the following are false.

(i)

(ii)

(iii)

(iv)

(v)

(vi)