Chapter 1.2, Problem 61HP

To determine

To explain: Commutative Property does not hold true for subtraction or division.

Explanation of Solution

For any real numbers the sets are tabulated below:

Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity	$a+0=0+a=a$ 0 is identity element for addition	$a\cdot 1=1\cdot a=a$ 1 is identity element for Multiplication
Inverse	$a+\left(-a\right)=\left(-a\right)+a=0$   $\left(-a\right)$ is inverse element for addition	$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1$   $\frac{1}{a}$ is inverse element for Multiplication
Closure	$a+b$ is real number	$a\cdot b$ is real number
Distributive	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$

Now, for subtraction let us consider two numbers 5 and 6.

  $\begin{array}{l}5-6=-1\\ 6-5=1\\ 5-6\ne 6-5\end{array}$

So, Commutative Property does not hold true for subtraction.

Now, for division let us consider two numbers 4 and 8.

  $\begin{array}{l}\frac{8}{4}=2\\ \frac{4}{8}=\frac{1}{2}\\ \frac{4}{8}\ne \frac{8}{4}\end{array}$

So, Commutative Property does not hold true for division.