**How to Identify Whether the Given Set is Finite or Infinite ?**

Here we are going to see how to check if the given set is finite or infinite.

**Finite set :**

If the number of elements in a set is zero or finite, then the set is called a finite set.

**Infinite set :**

A set is said to be an infinite set if the number of elements in the set is not finite.

**Question 1 :**

Write the set {−1, 1} in set builder form.

**Solution :**

{ x : x ∈ R, where x^{2} = 1 }

**Question 2 :**

State whether the following sets are finite or infinite.

(i) {x ∈ N : x is an even prime number}.

**Solution :**

First let us list out the elements in the given set. The set will contain even prime number.

2 is the only even prime number. The set is containing 1 element.

Hence the given set is finite.

(ii) {x ∈ N : x is an odd prime number}.

**Solution :**

First let us list out the elements in the given set. The set will contain odd prime number.

A = {3, 5, 7, 11, ..............}

Hence the given set is infinite set.

(iii) {x ∈ Z : x is even and less than 10}.

**Solution :**

Z means set of all positive and negative integers.

A = {-∞,................-1, 0, .......∞}

Hence the given set is infinite set.

(iv) {x ∈ R : x is a rational number}.

**Solution :**

R means real values, which is uncountable.

Hence the given set is known as infinite set.

(v) {x ∈ N : x is a rational number}.

**Solution :**

N means set of positive integers which is uncountable. Hence the given set is infinite set.

**Question 3 :**

By taking suitable sets A,B,C, verify the following results:

(i) A × (B ∩ C) = (A × B) ∩ (A × C).

**Solution :**

Let A = {1, 2, 3}, B = {1, 2} and C = {2, 3 4}

L.H.S

(B n C) = {2}

A × (B ∩ C) = {1, 2, 3} × {2}

A × (B ∩ C) = { (1, 2)(2, 2) (3, 2) } ------(1)

R.H.S

(A × B) = {1, 2, 3} × {1, 2}

(A × B) = {(1, 1) (1, 2) (2, 1) (2, 2)(3, 1) (3, 2)}

(A × C) = {1, 2, 3} × {2, 3 4}

(A × C)

= {(1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4)}

(A × B) ∩ (A × C) = { (1, 2)(2, 2) (3, 2) } ------(2)

(1) = (2)

Hence proved.

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