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.

Answer 1

Hint:From the given conditions, check if they contain elements or not. In an empty set there won’t be any value that will satisfy the condition.If there are values that satisfy the condition then they are not examples of empty sets.

Complete step-by-step answer:
The empty set is the unique set having no elements, its size or cardinality, which is the count of elements in the set is zero. We can ensure that the empty set exists by including an axiom of empty set. Now the empty set can be also referred to as the null set. However null set is a distinct notion within the context of measure theory.
Null or empty set can be also described as a set of measure zero. The common notations for the empty set include \['\left\{ {} \right\}'\] and\[\phi \].
Now for the given 2 conditions let us state whether they are examples of empty set or not.

Set of all even natural number divisible by 5.
We said then in an empty set there are no values that will satisfy the condition. Now, here we have even natural numbers like 10, 20, 30, 40, 50 etc. Which are divisible by 5. That is all natural numbers that end with zero are even and divisible by 5.
Set ‘S’ is the set containing all even natural numbers divisible by 5. Then,
\[S=\left\{ 10,20,30,40,50..... \right\}\]
Similarly, S is not an empty set. Hence it is not an example of an empty set.
Set of all even prime numbers.
So, we state out the prime numbers, we have 2, 3, 5, 7, 11, 13, 17, 19….. etc. of their prime number: 2 is an even prime number. If ‘S’ is the containing all even prime numbers, then,
\[S=\left\{ 2 \right\}\], clearly the set S is not an empty set.
Thus it is not an empty set.
Hence both the given statements are not examples of empty set. As they both have values that satisfy the given condition.

Note: We can give some examples of empty sets as
(i) The set of humans living on the Moon, which is an empty set.
(ii) The set of months with 32 days in a year.
(iii) The set of English words starting with \[x\] and ending with \[Q\] etc. are some examples of empty sets.