Question

Which of the following are countably infinite and uncountably infinite?
(i) Set of natural numbers
(ii) Set of real numbers
A) Both (i) and (ii) are uncountably infinite
B) (i) uncountably infinite and (ii) countably infinite
C) (i) countably infinite and (ii) uncountably infinite
D) Both (i) and (ii) are countably infinite

Answer 1

Here, we are asked to find which of the following is countably infinite and uncountably infinite.
Check for the natural numbers if we are able to predict the next number. If we are able to do so, then it is a countably infinite set or else it is an uncountably infinite set.
Similarly, Check for the real numbers if we are able to predict the next number. If we are able to do so, then it is a countably infinite set or else it is an uncountably infinite set.

Complete step by step solution:
Here, we are asked to find which of the following is countably infinite and uncountably infinite.
So, we will do so by taking help of some examples.
Let -4, -2, 0, 2, 4, ... is the sequence of numbers given to us.
Now, we can predict the next number of the sequence as the difference between two consecutive terms is 2. So, we can predict the elements in the above set which go up to infinite. Thus, it is a countably infinite set.
Similarly, the set of natural numbers is a set of numbers starting from 1 up to infinite with difference 1 between two consecutive terms.
So, natural numbers are a set of countably infinite elements.
Also, let 0, 5, 7, 10, 18, ... be the set given to us.
Here, we are not able to predict the next term in the sequence. So, we cannot predict the elements in the given sequence. Thus, it is an uncountably infinite set.
Similarly, the set of real numbers is an uncountably infinite set as we cannot predict the number that comes after any given number. For example, if we are asked to give a real number between 0 and 1, it will be difficult to give the exactly similar answer as the number asked, because there are infinitely many real numbers between 0 and 1.

So, option (C) is the correct answer.

Note:
Natural numbers:
Natural numbers are a set of numbers starting from 1 up to infinite. It is denoted by N. Thus,
N = \left\\{ {1,2,3,4,5,} \right\\}
Real numbers:
Real numbers are a set of all numbers. It is denoted by R. Thus, R includes all natural, whole, rational, irrational numbers and integers.
Also, $N \subset R$ .