Chapter 1.4, Problem 30P

Open-Ended Provide examples to show the following.

a. The associative property of addition holds true for negative integers.

b. The commutative property of multiplication holds true for non-integers.

c. The multiplicative property of negative one holds true regardless of the sign of the number on which the operation is performed.

d. The commutative property of multiplication holds true if one of the factors is zero.

To determine

To show: The Associative property for addition holds true for negative integers by providing example.

Explanation of Solution

Given information:

Associative property for addition-

  $a+\left(b+c\right)=\left(a+b\right)+c$

Example:-

Take ${\mathbb{Z}}^{-}$(set of negative integers),

-2,-5,-7$\in {\mathbb{Z}}^{-}$Then

Applying associative property,

  $\begin{array}{l}-2+\left(-5-7\right)\\ =-2-12\\ =-14\end{array}$

And

  $\begin{array}{l}(-2+(-5))-7\\ =(-2-5)-7\\ =-7-7\\ =-14\end{array}$

Thus, $a+\left(b+c\right)=\left(a+b\right)+c$

Similarly, it can be shown by taking any negative integers.

To determine

To show: the Commutative property of multiplication holds true for non integers by providing example.

Explanation of Solution

Given information:

Commutative property of multiplication: $a b=ba$

Example:-

Take Take $\mathbb{Q}$- $\mathbb{Z}$(set of non integers)

  $\frac{1}{2},\frac{2}{5}\in \mathbb{Q}$- $\mathbb{Z}$Then

Applying commutative property,

  $\frac{1}{2}\cdot \frac{2}{5}=\frac{1}{5}$

And

  $\frac{2}{5}\cdot \frac{1}{2}=\frac{1}{5}$

Thus, $ab=ba$

Similarly, it can be shown by taking any non integer.

To determine

To show: the multiplicative property of negative one holds true regardless of the sign of number on which the operation is performed for it provide example.

Explanation of Solution

Given information:

Multiplicative property of negative one:- $a\cdot (-1)=-a$

Example:-

Multiplicative property of negative one:- $r\cdot (-1)=-r$

Here, ‘ r’ can be any number,

So, $4\cdot (-1)=-4$

To determine

To show: - the commutative of multiplication holds true if one of the factors is zero for it provide an example.

Explanation of Solution

Given information:

Commutative property if one of factor is zero:- $a\cdot 0=0\cdot a$

Example:-

Commutative property if one of factor is zero:- $a\cdot 0=0\cdot \le a$

Take any number, it will hold true.

For example,

  $\begin{array}{l}4\cdot 0=0\cdot 4\\ 0=0\end{array}$Or

  $\begin{array}{l}2\ast 0=0\ast 2\\ 0=0\end{array}$