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) Both p and q are true
D) Both p and q are false

Answer 1

In the question, we have given that p: All integers are rational numbers and q: Every rational number is an integer
So, we will find which statement is correct
so, by the concept of rational number and integer we will find which statement is correct.

Complete step by step solution:
In the question, we have given that p: All integers are rational numbers and q: Every rational number is an integer
So, we have to find which statement is correct.
So, we must know that 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 true but we have to prove it.
$\because$Rational number is the number of the form $\dfrac{p}{q}$ where $q \ne 0$
So, let us take the example to prove the statements
$\because \dfrac{6}{7}$ is not an integer but still a rational number.
Now, to prove that the above example is not an integer, we have to find the decimal value of the number considered as:
$\because \dfrac{6}{7} = 0.85$
So, it gives the value 0.85 and by the definition of integers, it is not an integer value.
But when we go for the definition of rational number, $\dfrac{6}{7}$ is of the form $\dfrac{p}{q}$ and the denominator is also not zero which stated that it is a valid rational number.
$\therefore$ It proves that statement q is false.
Similarly, if we take any integer value let us suppose 4, so we can write it in the form: $\dfrac{4}{1}$
So, it gives the rational form as $\dfrac{p}{q}$ and its denominator is not zero
$\therefore$ It proves that statement p is true.
$\therefore$ statement p is true and statement q is false.

Note:
Here, this question refers to the concept of rational numbers. Which can be defined as “A number in form of $\dfrac{p}{q}$ where, p is called a numerator and q is called a denominator, also $q \ne 0$ ”
All integers and fractions are rational numbers.
The number 0 is neither a positive nor a negative rational number.