Question

Write the negation of: All natural numbers are integers and all integers are not natural numbers.
A.All natural numbers are not integers and all integers are not natural numbers.
B.All natural numbers are integers and all integers are natural numbers.
C.Some natural numbers are integers and all integers are not natural numbers.
D.All natural numbers are not integers and some integers are natural numbers.

Answer 1

Hint:
In this question, we are asked to write the negation of a given statement. To do that we will first identify the different components within the given statement. We will then write their negations separately. Lastly, we will integrate them and write them together to obtain the required statement.

Complete step by step answer:
The negation of a statement means to deny the statement completely. To do that we either write the exact opposite of the statement or we add negating words like, ‘no’, ‘not’ etc. with the subject matter in our statement to negate it. If there are already negating terms present, we remove them.
For example – The negation of the statement that Ram is taller will be that Ram is not taller.
Here, we have added a negating word to change the original meaning of the statement completely and to deny what was said.
Now, to write the negation of the statement - All natural numbers are integers and all integers are not natural numbers, we will first check for the different simple sentences in this compound statement.
There are two different simple sentences in this given statement. These are –
-All natural numbers are integers.
-All integers are not natural numbers.
We will now write the negation of these statements by adding the negating terms to them.
The negation of the first sentence will be –
All natural numbers are not integers.
The negation of the second sentence will be –
All integers are natural numbers.
We will now integrate the two negating sentences that we have obtained using ‘and’.
Hence, the required statement is –
All natural numbers are not integers and all integers are natural numbers.
As this statement resonates the most with option D amongst the provided choices, hence, the correct answer for the statement is option D.

Note:
We might commit an error in writing the negation for the simple sentence ‘all integers are not natural numbers’. This is because it already contains a negating word, so we may feel that it need not be changed. However, writing the negation of a statement means to completely deny it. So, in whatever way the sentence is presented to us, we have to prepare the exact opposite meaning or the denial of that. Negation does not mean to add only negating terms, to provide it a negative meaning.