Question

$\mathrm{What}\mathrm{can}\mathrm{be}\mathrm{said}\mathrm{about}\mathrm{the}\mathrm{following}\mathrm{statements}?\mathrm{Statement}1:\exists a\mathrm{natural}\mathrm{number}a\mathrm{such}\mathrm{that}{a}^2\le a.\mathrm{Statement}2:\forall \mathrm{rational}\mathrm{numbers}a,a=\frac{p}{q},\mathrm{where}p\mathrm{and}q\mathrm{are}\mathrm{integers}\mathrm{and}q\ne 0.$

• A. Statement 1 is true ,Statement 2 is false.
• B. Statement 1 is false ,Statement 2 is true.
• C. Statement 1 and 2 both are true.
• D. Statement 1 and 2 both are false.

Answer 1

The correct option is C Statement 1 and 2 both are true.

Statement 1 requires the existence of at least 1 natural number for which the given inequality is true.And it is evident that the inequality $a^2\le a$ is true for a=1.Hence, Statement 1 is true.
Statement 2 requires that all rational numbers can be expressed in the given form, which is the definition of rational numbers.