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Question: 12. Which of the following are examples of the null set ? (i) Set of odd natural numbers divisible b...

  1. Which of the following are examples of the null set ? (i) Set of odd natural numbers divisible by 2. (ii) Set of even prime numbers. (iii) {x:x is a natural number, x<5 and x>7}\{x: x \text{ is a natural number, } x < 5 \text{ and } x > 7\} (iv) {y:y is a point common to any two parallel lines}\{y: y \text{ is a point common to any two parallel lines}\}
A

Set of odd natural numbers divisible by 2.

B

Set of even prime numbers.

C

{x:x is a natural number, x<5 and x>7x: x \text{ is a natural number, } x < 5 \text{ and } x > 7}

D

{y:y is a point common to any two parallel linesy: y \text{ is a point common to any two parallel lines}}

Answer

The null sets are:

  • Set of odd natural numbers divisible by 2.
  • {x:x is a natural number, x<5 and x>7}\{x: x \text{ is a natural number, } x < 5 \text{ and } x > 7\}
  • {y:y is a point common to any two parallel lines}\{y: y \text{ is a point common to any two parallel lines}\}
Explanation

Solution

A null set (or empty set) is a set containing no elements.

  • (i) Set of odd natural numbers divisible by 2: Odd numbers are, by definition, not divisible by 2. Therefore, there are no odd natural numbers that are divisible by 2.

  • (ii) Set of even prime numbers: The number 2 is an even number and is also a prime number. Thus, the set contains the element {2}. This is not a null set.

  • (iii) {x:x is a natural number, x<5 and x>7}\{x: x \text{ is a natural number, } x < 5 \text{ and } x > 7\}: A natural number cannot be simultaneously less than 5 and greater than 7. There is no such number.

  • (iv) {y:y is a point common to any two parallel lines}\{y: y \text{ is a point common to any two parallel lines}\}: Parallel lines never intersect, meaning they have no common points.