Question

If A : The quotient of two integers is always a rational number, and\[\]
R : $\dfrac{1}{0}$ is not rational, then which of the following statements is true ?\[\]
A. A is true and R is the correct explanation of A.\[\]
B. A is false and R is the correct explanation of A. \[\]
C.A is true and R is false.\[\]
D. Both A and R are false.\[\]

Answer 1

Hint: We check truth value of the given assertion “If A : The quotient of two integers is always a rational number” by dividing any integer $a$ by any integer $b$. We check the truth value of reason “R : $\dfrac{1}{0}$ is not rational” by taking $a=1,b=0$. We check whether R is correctly explains truth value of A.\[\]

Complete step by step answer:
We know that an integer is a number without any fractional part which may be negative or positive. A rational number is a number that can be expressed in the form $\dfrac{p}{q}$ where $p,q$ integers are and $q$ must not be zero. $p$ is called numerator and $q$ is called denominator. \[\]
We are given an assertion-reason type question where the assertion is A : The quotient of two integers is always a rational number and the reason for the given assertion is R : $\dfrac{1}{0}$ is not rational.\[\]
We need to check first whether the assertion is true or not. We know when we divide an integer say $a$ by the integer say $b$ then we call $a$ the dividend and $b$ the divisor. We get quotient $q$where $q$ can be rational or integer after the division. Their relationship is
\[a=bq\]
If $q$is an integer the we call $b,q$ the factors of $a$, for example if we divide the integer $a=24$ by integer $b=4$ we get quotient as an integer $q=6$ which we can write as $24=4\times 6$ but when we divide 4 by 24 we get quotient $q=\dfrac{1}{6}$ , a rational number.\[\]
 If $q$ is an integer it can be expressed as a rational number with denominator $q$ but let us consider the case where $b=0$ if we want to divide any integer $a$ by $b=0$ we cannot because division by 0 is not defined. So the quotient for $\dfrac{a}{0}$does not exist and hence the quotient is not always rational. So A is false. \[\]
The reason given here that R: $\dfrac{1}{0}$ is not rational .If we assume $a=1$ the reason become true and also gives the correct explanation for falsity of A. So A is false and R is the correct expiation for it.\[\]

So, the correct answer is “Option B”.

Note: We note that $\dfrac{a}{0}$ is neither rational nor irrational, it is not defined. If we want only integral quotients then we shall get remainder $r$ when we divide $a$ by $b\ne 0$as $a=bq+r$. The numbers that cannot be expressed in the form $\dfrac{p}{q},q\ne 0$where are called irrational numbers. It is because we may not have integral quotients the set of integers are not closed under division.