Different Type of Sets - MathsTips.com

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) $X=\left\{1, 2, 3, 4\right\}$ , which has 4 members.

(ii) $Y= \left\{x \mid x \in \mathbb{W}, x \leq 9\right\}$ , which has 10 members.

Infinite Set:

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

Examples: (i) The set $\mathbb{W}$ of whole numbers.

(ii) $Y =\left\{ x \mid x \in \mathbb{Z}, x \leq 1 \right\}$

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

$\mathbb{W} =\left\{0, 1, 2,...\right\}$

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

$Y = \left\{ ..., -4, -3, -2, -1, 0, 1 \right\}$ .

Empty set:

A set which has no members is called an empty set or a null set. The empty set is denoted by $\phi$ or $\left\{\right\}$ .

Example: The set $\left\{x \mid 2x+1=0, x \in \mathbb{W} \right\}$ is an empty set because

$2x+1=0 \Rightarrow 2x= -1 \Rightarrow x= \dfrac {-1}{2} \notin \mathbb{W}$ .

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) $A=$ { $x \mid x$ is neither prime nor composite }. This is a singleton set containing one element, i.e. 1.

(ii) $B=$ { $x \mid x$ 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: $A= \left\{ x \mid x \in \mathbb{W} , x<2 \right\}$ . 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 $\xi$ .

Example: If the sets involved in a discussion are sets of some natural numbers then the set $\mathbb{N}$ 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 $n(A)$ . The cardinal number of the empty set $\phi$ is 0 because $\phi$ has 0 members. So, $n(\phi)=0$ . And the cardinal number of an infinite set cannot be found because such a set has countless members.

Examples: (i) If $A= \left\{ -3, -2, -1, 0, 1 \right\}$ then $n(A)=5$ .

(ii) If $A=$ { $x \mid x$ is a letter of the word PATNA} then $n(A)=4$ because A in the tabular form is $\left\{P, A, T, N \right\}$ .

Note: If $n(A)=1$ , we call set A a singleton set.

If $n(A)=2$ , 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 $A \leftrightarrow B$ and read this as “A is equivalent to B”.

$A \leftrightarrow B$ if $n(A)=n(B)$ .

Examples: Let $X= \left\{ 0, 2, 4 \right\}$ , and $Y=$ { $x \mid x$ is a letter of the word DOOR} .

Then, $n(X)=3$ and $n(Y)=3$ because $B=\left\{ D, O, R \right\}$ . So $X \leftrightarrow Y$ .

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 $A \subseteq B$ . 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 $B \supseteq A$ .

Example: Let $A=\left\{ 1, 2 \right\}, B=\left\{ 1, 3 \right\}$ and $C=\left\{ 1, 2, 4 \right\}$

Then, $1 \in A$ and $1 \in B$ . But $2 \in A$ and $2 \notin B$ . So, $A \nsubseteq B$ .

Similarly, $B \nsubseteq C$ , and $C \nsubseteq A$ .

Now, $1 \in a$ and $1 \in C$ . Also, $2 \in A$ and $2 \in C$ . Thus, all the members of A are members of C. So, $A \subseteq C$ . Also, $C \supseteq A$ .

Note: (i) Since the empty set $\phi$ 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., $A \subseteq A$ .

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 $A \subseteq B$ and $B \subseteq A$ . This is denoted as $A=B$

Example: Let $A= \left\{ 1, 2, 3, 4 \right\}$ and $B=\left\{ x \mid x <5, x \in \mathbb{N} \right\}$

Writing in the tabular form, $B=\left\{ 1, 2, 3, 4 \right\}$

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

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

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

Exercise:

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

(i) $A= \left\{ x \mid x \in \mathbb{N}, -1<x<3 \right\}$

(ii) $B=$ { $x \mid x$ is a prime number of digit one}

(iii) $C=$ { $x \mid x$ is a two-digit number divisible by 15}

(iv) $D=$ { $x \mid x$ is a letter of the word BHARATI}

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

(i) $\left\{ a, e, I, o, u \right\}$

(ii) $\left\{ 0, 5, 10, 15 \right\}$

(iii) $\left\{ 1, 3, 5, 7,... \right\}$

(iv) $\left\{ ...,-3, -2, -1, 0 \right\}$

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

(i) $A= \left\{ x \mid x>4, x \in \mathbb{N} \right\}$

(ii) $B= \left\{ x \mid x<5, x \in \mathbb{Z} \right\}$

(iii) $C= \left\{ x \mid x<5, x \in \mathbb{W} \right\}$

(iv) $\left\{ ...,-3, -2, -1, 0 \right\}$

(v) $E= \left\{ x \mid x<1, x \in \mathbb{N} \right\}$

(vi) $F= \left\{ x \mid x =\dfrac{1}{n^2+1}, n \in \mathbb{Z} \right\}$

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

(i) $A=\left\{ 0 \right\}$

(ii) $= \left\{ x \mid x^2-9=0, x \in \mathbb{Z} \right\}$

(iii) $= \left\{ x \mid x^2=25, x \in \mathbb{N} \right\}$

(iv) $= \left\{ x \mid 5x-4=0, x \in \mathbb{W} \right\}$

5. Let $X=$ { $x \mid x$ is a letter of the word MINISTER}

And $Y=$ { $x \mid x$ is a letter of the word SINISTER}

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

(i) $X \subseteq Y$

(ii) $Y \subseteq X$

(iii) $X=Y$

(iv) $X \leftrightarrow Y$

6. Let $P= \left\{ -1, 1 \right\}, Q= \left\{ x \mid x^2=4, x \in Z \right\}, R=\left\{ -2, -1, 1 \right\}$ . State which of the following are false.

(i) $P \subseteq Q$

(ii) $P \subseteq R$

(iii) $Q \subseteq R$

(iv) $P \leftrightarrow Q$

(v) $Q \leftrightarrow R$

(vi) $P=Q$