Question

Which of the following statements is not true?
A) Every whole number has a successor.
B) Every whole number has a predecessor.
C) 0 is the least whole number.
D) Every natural number is a whole number.

Answer 1

Hint: We will first understand what is meant by the terms “successor” and “predecessor”. After that, we will inspect each of the given options one by one for its truth value. We will thus have the required answer.

Complete step-by-step answer:
Let us first write the definitions of the terms “successor” and “predecessor”.
Successor: Successor means the next term of a particular term. If we need to find a successor of a whole number, we just need to add ‘1’ to it.
Predecessor: Predecessor means the term previous to a particular term. If we need to find a predecessor of a whole number, we just need to subtract 1 from it.
Now, let us go through all the options one by one:
Option A: Every whole number has a successor
If we add 1 to any of the whole numbers, we will land up to a whole number.
The set of whole numbers = {0, 1, 2, ……….}
The set of its successors = {0 + 1, 1 + 1, 2 + 1, …………..} = {1, 2, 3, …….}
Therefore, the set of successors of whole numbers is $\mathbb{N}$.
Hence, every whole number is a successor.
Hence, option (A) is true.
Option B: Every whole number has a predecessor
If we subtract 1 from any of the whole number, we will get the predecessor.
The set of whole numbers = {0, 1, 2, ……….}
The set of its successors = {0 - 1, 1 - 1, 2 - 1, …………..} = {-1, 0, 1, …….}
But we see that -1 does not belong to the set of whole numbers.
Hence, option (B) is false.
Option C: 0 is the least whole number
We know that the set of whole numbers is given by {0, 1, 2,……..}
We see that 0 < 1 < 2 < 3 < ……………. and so on.
Therefore, option (C) is true.
Option D: Every natural number is a whole number
We know that $\mathbb{N}=\{1,2,3,.......\}$ and we can clearly see that whole numbers have a zero extra to its set. Therefore, the set of natural numbers is a subset of whole numbers.

Hence, option (D) is correct.

Note: The students must note that the statement B which was found to be incorrect was not technically incorrect because every whole number does have a predecessor because 0 has a predecessor of -1 but it is just that it does not lie in whole numbers only like all the successors lie in whole numbers only. Therefore, we termed B to be incorrect.