Which of the following are examples of null set?

{ x : x is a natural numbers, x < 5 and x > 7 }

Solution

Let $space space space space space space space space space straight D space space equals space space left curly bracket space straight x space colon space straight x space element of space bold N bold comma bold space bold x bold space bold less than bold space bold 5$ and simultaneously x > 7}
There is no natural number which is simultaneously < 5 and > 7.
$rightwards double arrow$ $space space space space space space space space straight D space space equals space open curly brackets space space close curly brackets space equals space straight ϕ$
Hence, D is a null set.