Question

Write 'T' for true and 'F' for false for each of the following:
(i) Rational numbers are always closed under subtraction.
(ii) Rational numbers are aways closed under division.
(iii) 1 ÷ 0 = 0.
(iv) Subtraction is commutative on rational numbers.
(v) $-\left(\frac{-7}{8}\right)=\frac{7}{8}.$

Answer 1

(i) T

​If $\frac{\mathit{a}}{\mathit{b}}\mathrm{and}\mathit{}\frac{\mathit{c}}{\mathit{d}}$ are rational numbers, then $\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$ is also a rational number because $ad,bc\mathrm{and}bd$ are all rational numbers.

(ii) F

​Rational numbers are not always closed under division. They are closed under division only if the denominator is non-zero.

(iii) F

$1÷0$cannot be defined.
​
(iv) F

​Let $\frac{a}{b}\mathrm{and}\frac{c}{d}$ represent rational numbers.

Now, we have:

$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$
$\frac{c}{d}-\frac{a}{b}=\frac{bc-ad}{bd}$

∴ $\frac{a}{b}-\frac{c}{d}\ne \frac{c}{d}-\frac{a}{b}$

(v) T
`
$-\left(\frac{-7}{8}\right)=-1\times \left(\frac{-7}{8}\right)=\frac{-1\times -7}{8}=\frac{7}{8}$