Write ‘T’ for true and ‘F’ for false for each of the following:
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) $-\left(\frac{-7}{8}\right)=\frac{7}{8}$

Solution:

(i) T

If $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers, then $\frac{a}{b}-\frac{c}{d}=\frac{a d-b c}{b d}$ is also a rational number because $a d, b c$ and $b d$ 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

$1 \div 0$ cannot be defined.

(iv) F

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

Now, We have:

$\frac{a}{b}-\frac{c}{d}=\frac{a d-b c}{b d}$

$\frac{c}{d}-\frac{a}{b}=\frac{b c-a d}{b d}$

$\therefore \frac{a}{b}-\frac{c}{d} \neq \frac{c}{d}-\frac{a}{b}$

(v) T

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

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