State whether each of the following statements is true or false:

(i) The difference of two rational numbers is always a rational number.

(ii) $\dfrac{2}{3} - \dfrac{3}{5} = \dfrac{3}{5} - \dfrac{2}{3}$

(iii) 1 is the additive identity for rational numbers.

(iv) Subtraction is commutative on rational numbers.

(v) There exists a rational number which is its own additive inverse.

(vi) $\dfrac{19}{-5} + \dfrac{-3}{11} = \dfrac{19}{5} + \dfrac{3}{11}$

(i) Rational numbers are closed under subtraction, i.e. if $\dfrac{a}{b} \text{ and } \dfrac{c}{d}$ are two rational numbers, then $\dfrac{a}{b} - \dfrac{c}{d}$ is also a rational number.

Hence, the given statement is true.

(ii) Solving L.H.S.,

$\dfrac{2}{3} - \dfrac{3}{5} = \dfrac{2 \times 5 - 3 \times 3}{15} = \dfrac{10 - 9}{15} = \dfrac{1}{15}$

Solving R.H.S.,

$\dfrac{3}{5} - \dfrac{2}{3} = \dfrac{3 \times 3 - 2 \times 5}{15} = \dfrac{9 - 10}{15} = \dfrac{-1}{15}$

Clearly, L.H.S. ≠ R.H.S.

Hence, the given statement is false.

(iii) For every rational number $\dfrac{a}{b}$, we have $\dfrac{a}{b} + 0 = \dfrac{a}{b} = 0 + \dfrac{a}{b}$, so 0 is the additive identity and 1 is the multiplicative identity.

Hence, the given statement is false.

(iv) For any two rational numbers $\dfrac{a}{b} \text{ and } \dfrac{c}{d}$, we have $\dfrac{a}{b} - \dfrac{c}{d} ≠ \dfrac{c}{d} - \dfrac{a}{b}$. For example, $\dfrac{2}{3} - \dfrac{3}{5} = \dfrac{1}{15}$ but $\dfrac{3}{5} - \dfrac{2}{3} = \dfrac{-1}{15}$.

Hence, the given statement is false.

(v) The additive inverse of the rational number 0 is 0 itself, as 0 + 0 = 0.

Hence, the given statement is true.

(vi) Solving L.H.S.,

$\Rightarrow \dfrac{19}{-5} + \dfrac{-3}{11} \\[1em] \Rightarrow \dfrac{-19}{5} + \dfrac{-3}{11} \\[1em] \Rightarrow \dfrac{(-19) \times 11 + (-3) \times 5}{55} = \dfrac{-209 - 15}{55} = \dfrac{-224}{55}$

Solving R.H.S.,

$\Rightarrow \dfrac{19}{5} + \dfrac{3}{11} \\[1em] \Rightarrow \dfrac{19 \times 11 + 3 \times 5}{55} = \dfrac{209 + 15}{55} = \dfrac{224}{55}$

Clearly, L.H.S. ≠ R.H.S.

Hence, the given statement is false.