Math, asked by arulmozhi4948, 10 months ago

With the help of a suitable example show that the subtraction is non -commutation and non-associated for integers

Answers

Answered by hukam0685
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,one can say that subtraction is non-commutative and non-associative.

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

Hope it helps you.

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