Chapter R, Problem 1RE

Complete the table.

To determine

To fill: The table given below as,

Answer to Problem 1RE

The complete table is,

Explanation of Solution

It is observed that a parenthesis “ $\left.\right)$ � or “ $\left(\right.$ � indicates that an endpoint is not included in an interval. A bracket “ $\left.\right]$ � or “ $\left[\right.$ � indicates that an endpoint is included in the interval. The real number line extends infinitely far to the left and right.

(a)

Consider the given number line.

It is seen that an open bracket “ $\left[\right.$ �appears at $-3$ on the number line. Thus, the interval starts from this point including it. A closed parenthesis “ $\left.\right)$ � is set at $7$ on the number line. Thus, the interval ends at this point and this point will be included in the interval of the set function.

Write the set function in the interval notation for the given number line.

$\left[-3,7\right)$

Write the set function in the set-builder notation for the given number line.

$\left\{x\left|x<7,x\ge -3\right.\right\}$

Therefore, the set function in the interval notation is $\left[-3,7\right)$ and the set function in the set-builder notation is $\left\{x\left|-3\le x<7\right.\right\}$ .

(b)

Consider the given set in the interval notation.

$\left(2.1,\infty \right)$

Here, the set function starts from the point $2.1$ . As the open parenthesis is used, the starting point, that is, $2.1$ will be excluded from the set function. The set function includes all the real numbers till infinity.

Draw the number line for the given set of interval notation.

Write the set function in the set-builder notation for the given set.

$\left\{x\left|x>2.1\right.\right\}$

Therefore, the set function in the set-builder notation is $\left\{x\left|x>2.1\right.\right\}$ .

(c)

Consider the given set in the set-builder notation.

$\left\{x\left|4\ge x\right.\right\}$

As seen from the set-builder notation, the variable $x$ is greater than or equal to $4$ . Thus, the set function contains all the real numbers greater than or equal to $4$ .

For the inclusion of value $4$ , an open bracket “ $\left[\right.$ � will be used on the number line and interval notation. The set function will continue until infinity, so a closed parenthesis “ $\left.\right)$ � will be used.

Draw the number line diagram for the given set-interval notation.

Write the set function in the interval notation for the given set-builder notation.

$\left\{x\left|4\ge x\right.\right\}$

Therefore, the set function in the interval notation is $\left(-\infty ,4\right]$ .

Hence, the complete table is,