Chapter P.2, Problem 58E

Solution

The given statement that a vector is an element of vector space is true or false.

True

Given information:

Statement given is: “If a is a real number than $\left|a\right|\ge 0$ .”

Formula used:

For any x :

  $\left|x\right|=\left\{\begin{array}{l}x\text{ if }x\text{ is positive}\\ -x\text{ if }x\text{ is negative}\end{array}\right.$

Calculation:

In order to check whether the statement is true or false

“If a is a real number than $\left|a\right|\ge 0$ .” Use the definition of absolute value symbol for any x :

  $\left|x\right|=\left\{\begin{array}{l}x\text{ if }x\text{ is positive}\\ -x\text{ if }x\text{ is negative}\end{array}\right.$

Now since a is any real number, than if we take a to be negative number, i.e. $a<0$ then according to above

  $\left|a\right|=-a$

i.e.

  $\begin{array}{l}-a>0\\ \left|a\right|>0\end{array}$

If we take a to be positive number, i.e. $a>0$ then according to above

  $\left|a\right|=a>0$

And in case a is 0

  $\left|a\right|=0$

Thus, for any real number a $\left|a\right|\ge 0$ .

Thus, the given statement “If a is a real number than $\left|a\right|\ge 0$ ” is true.