Question

Which of the following are examples of empty set?
(i) \[\left\{ x:{{x}^{2}}-2=0\text{ and }x\text{ is rational} \right\}\]
(ii) \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number,\text{ }x<8\text{ } and \text{ simultaneously }x>12 \right\}\]

Answer 1

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

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 a 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 relations 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.
For the 1st statement, the condition was given as.
\[\left\{ x:{{x}^{2}}-2=0\text{ }and\text{ }x\text{ }is\text{ }rational \right\}\], then it will be an empty set as \[x=\pm \sqrt{2}\], is irrational number which does not satisfies the given condition means there is no element present in set according to given condition.Thus it is said to be empty set.
Similar if the 2nd statement, the condition was given as,
 \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number,\text{ }x<8\text{ }and\text{ }-2=0\text{ }and\text{ simultaneously }x>12 \right\}\], then
Natural no: greater than \[8\left( x<8 \right)=7,6,5,4,3,2,\]
Natural no: less than \[12\left( x>12 \right)=13,14,15,16,17,18,...\]
We don't have any common elements which satisfies the condition.Thus we can say that the given set is 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.