Chapter 1.4, Problem 26P

Writing Explain why the commutative and associative properties don’t hold true for subtraction and division but the identity properties do.

To determine

To explain:why the commutative and associative properties don't hold true for subtraction and division but the identity properties do.

Explanation of Solution

First we explain for subtraction

as we know that $a-b\ne b-a$ , provided $(a\ne b)$

In $a-b$ we subtract $b$ from $a$ to get result and in $b-a$ we subtract $a$ from $b$ to get result

For example: Let $a=2\&b=3$ , then

  $a-b=2-3=-1$ and

  $b-a=3-2=1$

Hence, $a-b\ne b-a$ so it is not commutative

hence not associative

Now we explain for division

as we know that $\frac{a}{b}\ne \frac{b}{a}$ , provided $(a=b\ne 0)$

In $\frac{a}{b}$ , we divide $a$ by $b$ to get result and in $\frac{b}{a}$ , we divide $b$ by $a$ to get result.

For example: Let $a=2\&b=3$ , then

  $\frac{2}{3}=0.667$ and

  $\frac{3}{2}=1.5$

Hence, $\frac{2}{3}\ne \frac{3}{2}$

so $\frac{a}{b}\ne \frac{b}{a}$ , so it is not commutative

hence not Associative

But they follow the identity property

Addition and subtraction identity is '0'

Let $a=2$ , so

  $a-0=a$

here $4-0=4$

Hence, they follow the identity property of subtraction.

Multiplication and division identity is '1'

Let $a=2$ , so

  $a*1=a$

here $2*1=2$

Hence they follow the identity property of division.