Question

4. Which of the following sets are empty sets ?

(i) {x: x² = 2 and x is a rational number} (ii) {x: x is an integer neither positive nor negative} (iii) {x: x is a boy student in a girls college} (iv) {x: x < 5 and also x > 5} (v) {x: x is a real number x² < 0}.

5. State, which of the following sets are finite and which infinite : (i) {x: x ∈ N and x is prime} (ii) {x: x is a positive integral root of x² - 2x - 15 = 0} (iii) {x: x is a human being in the world} (iv) {x: x is a multiple of 3} (v) {x: x is a real number between 1 and 2}

Answer 1

4. Empty Sets:

The sets that are empty sets are: (i) {x: x² = 2 and x is a rational number} (iii) {x: x is a boy student in a girls college} (iv) {x: x < 5 and also x > 5} (v) {x: x is a real number x² < 0}

5. Finite and Infinite Sets: (i) {x: x ∈ N and x is prime} - Infinite (ii) {x: x is a positive integral root of x² - 2x - 15 = 0} - Finite (iii) {x: x is a human being in the world} - Finite (iv) {x: x is a multiple of 3} - Infinite (v) {x: x is a real number between 1 and 2} - Infinite

Answer 2

Part 1: Identifying Empty Sets

An empty set is a set that contains no elements.

(i) {x: x² = 2 and x is a rational number}

If x² = 2, then x = $\pm\sqrt{2}$. Since $\sqrt{2}$ is an irrational number, there is no rational number x such that x² = 2.
Therefore, this set is an empty set.

(ii) {x: x is an integer neither positive nor negative}

The integer that is neither positive nor negative is 0. So, this set is {0}, which contains one element.
Therefore, this set is not an empty set.

(iii) {x: x is a boy student in a girls college}

By definition, a girls' college admits only girl students. Thus, there cannot be any boy student in a girls' college.
Therefore, this set is an empty set.

(iv) {x: x < 5 and also x > 5}

There is no number that can be simultaneously less than 5 and greater than 5. This condition is contradictory.
Therefore, this set is an empty set.

(v) {x: x is a real number x² < 0}

For any real number x, its square (x²) is always greater than or equal to zero (x² ≥ 0). There is no real number whose square is negative.
Therefore, this set is an empty set.

Sets that are empty sets: (i), (iii), (iv), (v)

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Part 2: Identifying Finite and Infinite Sets

A finite set has a countable number of elements, while an infinite set has an uncountable or unending number of elements.

(i) {x: x ∈ N and x is prime}

This set represents the set of all prime numbers {2, 3, 5, 7, 11, ...}. There are infinitely many prime numbers.
Therefore, this set is an infinite set.

(ii) {x: x is a positive integral root of x² - 2x - 15 = 0}

First, solve the quadratic equation x² - 2x - 15 = 0:
(x - 5)(x + 3) = 0
The roots are x = 5 and x = -3.
We are looking for positive integral roots. The only positive integral root is 5.
So, the set is {5}, which contains only one element.
Therefore, this set is a finite set.

(iii) {x: x is a human being in the world}

The number of human beings in the world, while very large, is a specific, countable number at any given moment. It is not infinite.
Therefore, this set is a finite set.

(iv) {x: x is a multiple of 3}

This set represents the multiples of 3: {3, 6, 9, 12, ...}. There are infinitely many multiples of 3.
Therefore, this set is an infinite set.

(v) {x: x is a real number between 1 and 2}

Between any two distinct real numbers, there are infinitely many real numbers. For example, 1.1, 1.01, 1.001, 1.5, 1.99, etc., are all real numbers between 1 and 2.
Therefore, this set is an infinite set.

Sets that are finite: (ii), (iii)
Sets that are infinite: (i), (iv), (v)