QUESTION

# The set of positive integers is ………………….A. InfiniteB. FiniteC. SubsetD. Empty

Hint: To solve this question we will use the basic concept of sets and functions. We will solve the question by assuming N to be the greatest positive integer such that no integer is greater than N and solve accordingly.

We know that 1 is a positive integer.
Now adding 1 unit to 1 we get,
$\Rightarrow 1 + 1 = 2$,
Which will always give,
$\Rightarrow 2 > 1$.
Similarly, let us assume N be the greatest positive integer such that no other positive integer is greater than N,
Now add 1 unit to N,
We get $N + 1$,
Since, we assumed that N is the greatest positive integer,
So, N should be greater than $N + 1$
$\Rightarrow N > N + 1$
Now subtracting N from both the sides we get,
$\Rightarrow N - N > N + 1 - N$
$\Rightarrow 0 > 1$
This is clearly false as 0 can never be greater than 1,
So, our assumption that N is the greatest positive integer such that no other positive integer is greater than N must be false.
$\Rightarrow N < N + 1$
Now adding 1 to both the sides we get,
$\Rightarrow N + 1 < N + 2$
Similarly,
$\Rightarrow N + 2 < N + 3$ and so on,
Now this will go on till infinity making an infinite series.
Therefore, we can conclude that there are infinitely many positive integers.
Hence, the correct answer is option (A) Infinite.

Note: The basic problem faced in the above question is how to approach the question as there are many ways. The above question can also be solved by the basic concept of function that is mapping or by the concept of cardinality of sets.