Question
'Rational numbers are commutative under addition but not commutative under subtraction.' Justify the statement with an example.
✓
Answer
Let $\frac{1}{2}$ and $\frac{1}{4}$ be two rational numbers.
Now, $\frac{1}{2}+\frac{1}{4}=\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$ which implies that rational numbers are commutative under addition.
Now, $\quad \frac{1}{2}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}$
But $\quad \frac{1}{4}-\frac{1}{2}=\frac{1-2}{4}=-\frac{1}{4}$
Thus, $\quad \frac{1}{2}-\frac{1}{4} \neq \frac{1}{4}-\frac{1}{2}$
So, rational numbers are not commutative under subtraction.
Now, $\frac{1}{2}+\frac{1}{4}=\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$ which implies that rational numbers are commutative under addition.
Now, $\quad \frac{1}{2}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}$
But $\quad \frac{1}{4}-\frac{1}{2}=\frac{1-2}{4}=-\frac{1}{4}$
Thus, $\quad \frac{1}{2}-\frac{1}{4} \neq \frac{1}{4}-\frac{1}{2}$
So, rational numbers are not commutative under subtraction.
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