Chapter 1.1, Problem 43E

(a)

To determine

To find: the set $B\cup C$ .

Answer to Problem 43E

  $B\cup C=\left\{x|x\le 5\right\}$

Explanation of Solution

Given information:

Three sets are givenas $A=\left\{x|x\ge -2\right\},B=\left\{x|x<4\right\}$ , and $C=\left\{x|-1<x\le 5\right\}$

Formula used:

For any two or more sets, union of the sets gives the elements which are in the either one of the sets and the all elements of all sets. Common elements are considered once. Union is represented by the symbol $\cup$ .

For any two or more sets, intersection of the sets gives the elements which are the only common elements of the sets. Intersection is represented by the symbol $\cap$ .

Calculation:

Consider the given sets.

  $B=\left\{x|x<4\right\}$

and

  $C=\left\{x|-1<x\le 5\right\}$

Now, $B\cup C$ will contains elements of B and C. Common elements will be considered once.

  $\begin{array}{c}B\cup C=\left\{x|x<4\right\}\cup \left\{x|-1<x\le 5\right\}\\ =\left\{x|x\le 5\right\}\end{array}$

(b)

To determine

To find: the set $B\cap C$ .

Answer to Problem 43E

  $B\cap C=\left\{x|-1<x<4\right\}$

Explanation of Solution

Given information:

Three sets are given as $A=\left\{x|x\ge -2\right\},B=\left\{x|x<4\right\}$ , and $C=\left\{x|-1<x\le 5\right\}$

Formula used:

For any two or more sets, union of the sets gives the elements which are in the either one of the sets and the all elements of all sets. Common elements are considered once. Union is represented by the symbol $\cup$ .

For any two or more sets, intersection of the sets gives the elements which are the only common elements of the sets. Intersection is represented by the symbol $\cap$ .

Calculation:

Consider the given sets.

and

  $C=\left\{x|-1<x\le 5\right\}$

Now, $B\cap C$ will contains common elements of B and C.

  $\begin{array}{c}B\cap C=\left\{x|x<4\right\}\cap \left\{x|-1<x\le 5\right\}\\ =\left\{x|-1<x<4\right\}\end{array}$