Question

Which of the following are examples of empty set?
(i) Set of all even natural numbers divisible by 5.
(ii) Set of all even prime numbers.
(iii) {$x:{x^2} - 2 = 0$ and $x$ is rational }
(iv) {$x$:$x$ is a natural number, $x < 8$ and simultaneously $x > 12$.
(v) {$x$:$x$ is a point common to any two parallel lines }

Answer 1

Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set. For each of the options first we will find the sets for each of the cases and later see if it is an empty set or not.

Complete step-by-step solution:
Empty set is a set which does not contain any element.
(i) Set of all even natural numbers divisible by 5.
Since, all numbers ending with 0. Here, except 0 is divisible by 5 and is an even natural number. Thus it is not an example of empty set.
(ii) Set of all even prime numbers.
Since, 2 is a prime number and is even, and it is the only prime which is even.
Therefore this is not an example of the empty set.
(iii) {$x:{x^2} - 2 = 0$ and $x$ is rational }
Here,${x^2} - 2 = 0\,\,\, \Rightarrow {x^2} = 2\,\, \Rightarrow x = \pm \sqrt 2 \,\varepsilon \,N$. There is no natural number whose square is 2.
Therefore it is an example of empty set.
(iv) {$x$:$x$ is a natural number, $x < 8$ and simultaneously $x > 12$.
There is no natural number less than 8 and greater than 12.
Thus it is an example of the empty set.
(v) {$x$:$x$ is a point common to any two parallel lines }
Here, no two parallel lines intersect at each other.
Thus it is an example of empty set.

Note: A set with no elements is called as an empty set. Empty set is also called a null set. Empty set is a subset of each and every set.