Question

If p:All integers are rational numbers and q: Every rational number is an integer, then which of the following statements is correct?
(a) p is False and q is True
(b) p is True and q is False
(c) Bothe p and q are True
(d) Both p and q are false

Answer 1

Hint: First, before that, we all should know what a rational number as a rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ . Similarly, an integer is the number that has an integer value and always comes in the total range of numbers from $ -\infty $ to $ \infty $ . Then, by taking appropriate examples for both the cases ,we can prove which statement is True and which statement is False.

Complete step-by-step answer:
In this question, we are supposed to find which of the given statements as p:All integers are rational numbers and q: Every rational number is an integer is correct.
So, before proceeding for this, we must know the condition that the rational number is always an integer is not necessary but an integer can be written as a rational number always.
Here, we know that the above condition is correct but we need to prove it before concluding it.
But, before that, we all should know what is a rational number.
So, the rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ .
However, an integer is the number that has an integer value and always comes in total range of numbers from $ -\infty $ to $ \infty $ .
Now, to take an example of any rational number which is not an integer is:
 $ \dfrac{3}{4} $
Here, the example considered by us as $ \dfrac{3}{4} $ is not an integer but still a rational number.
Now, to prove that it is not an integer, find the decimal value of the number considered as:
 $ \dfrac{3}{4}=0.75 $
So, it gives the value 0.75 and by the definition of integers, it is not an integer value.
But when we go for the definition of rational number, $ \dfrac{3}{4} $ is of the form $ \dfrac{p}{q} $ and also its denominator is not zero which states that it is a valid rational number proves q statement is False.
Subsequently, if we take any integer value let us suppose 2, so we can write it in the form as:
 $ \dfrac{2}{1} $
So, it gives the rational form as $ \dfrac{p}{q} $ and its denominator is not zero proves the statement p is True.
So, the correct answer is “Option B”.

Note: Now, to solve these types of questions, it was not necessary to take this fixed example as $ \dfrac{3}{4} $ to prove that a rational number is not necessarily an integer. So, in mathematics of the numbers, we have a number of rational numbers like $ \dfrac{1}{2},\dfrac{6}{8},\dfrac{1}{5} $ and much more that are not integers. Similarly, we can take any integer like 3, 4, -1 in the rational form to prove that every integer is a rational number.