Question

Explain by giving example properties of the group I not followed in subtraction of rational numbers

Answer 1

Step-by-step explanation:

Given that:Properties of the group are not followed in subtraction of rational number explain by giving example.

Solution:

Following are the properties which are not followed by subtraction of rational numbers

1) Commutative property:

It states that

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

But this is not followed by subtraction

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

Example: Let

$\frac{a}{b} = \frac{2}{3} \\ \\ \frac{c}{d} = \frac{5}{2} \\ \\ \frac{2}{3} - \frac{5}{2} = \frac{4 - 15}{6} \\ \\ = \frac{ - 11}{6} \\ \\ \frac{5}{2} + \frac{2}{3} = \frac{15 - 4}{6} \\ \\ = \frac{11}{6} \\ \\ \frac{ - 11}{6} \neq \frac{11}{6}$

Thus Commutative property is not followed by subtraction of two rational number.

2) Associative property:

Associative property is also not followed by subtraction of rational numbers.

$\frac{a}{b} + ( \frac{c}{d} + \frac{e}{f} ) =( \frac{a}{b} + \frac{c}{d} ) + \frac{e}{f}$

But it is not true for subtraction

$\bold{\frac{a}{b} - ( \frac{c}{d} - \frac{e}{f} ) \neq( \frac{a}{b} - \frac{c}{d} ) - \frac{e}{f}}$

Example:

Let

$\frac{a}{b} = \frac{2}{3} \\ \\ \frac{c}{d} = \frac{5}{6} \\ \\ \frac{e}{f} = \frac{7}{3} \\ \\ \frac{a}{b} - ( \frac{c}{d} - \frac{e}{f} ) = > \\ \\ \frac{2}{3} - ( \frac{5}{6} - \frac{7}{3}) = \frac{2}{3} - ( \frac{5 - 14}{6} ) \\ \\ = \frac{2}{3} - ( \frac{ - 9}{6}) \\ \\ = \frac{2}{3} + \frac{9}{6} \\ \\ = \frac{6 + 9}{6} = \frac{15}{6} \\ \\ \frac{2}{3} - ( \frac{5}{6} - \frac{7}{3}) = \frac{5}{2} \\ \\ ( \frac{a}{b} - \frac{c}{d} ) - \frac{e}{f} = ( \frac{2}{3} - \frac{5}{6} ) - \frac{7}{3} \\ \\ = ( \frac{4 - 5}{6} ) - \frac{7}{3} \\ \\ = \frac{ - 1}{6} - \frac{7}{3} \\ \\ = \frac{ - 1 - 14}{6} \\ \\ = \frac{ - 15}{6} = \frac{ - 5}{2} \\ \\ ( \frac{2}{3} - \frac{5}{6} ) - \frac{7}{3} = \frac{ - 5}{2} \\ \\ \frac{ - 5}{2} \neq \frac{5}{2}$

Thus,associative property is not followed by subtraction of rational numbers.

By this way one can say that Associative property and Commutative are not followed by subtraction of rational numbers.

Hope it helps you.