Chapter 1.3, Problem 33WE

(a)

To determine

To explain:

The set of real numbers is closed under addition or not.

Answer to Problem 33WE

The set is not closed under addition.

Explanation of Solution

The members of the set are real numbers greater than $-1$ and less than $1$ . As any two members of the set can include the same number used twice, choose the members, $\frac{2}{3}$ , and $\frac{2}{3}$ , find their sum.

  $\frac{2}{3}+\frac{2}{3}=\frac{4}{3}$

The numbers $\frac{4}{3}$ is greater than $1$ , which is not in the set.

Thus, the set is not closed under addition.

(b)

To determine

To explain:

The set of real numbers is closed under multiplication or not.

Answer to Problem 33WE

The set is closed under multiplication.

Explanation of Solution

Choose any two members from the set, $\frac{1}{2}$ , and $\frac{2}{3}$ , find their product.

  $\frac{1}{2}\times \frac{2}{3}=\frac{2}{6}$

Similarly, choose $\frac{3}{4}$ , and $\frac{1}{2}$ , find their product.

  $\frac{3}{4}\times \frac{1}{2}=\frac{3}{8}$

The numbers $\frac{2}{6}$ and $\frac{3}{8}$ are real numbers greater than $-1$ and less than $1$ , so they will be members of the set.

Therefore, the set is closed under multiplication.