Chapter 2.2, Problem 1E

a.

To determine

To find $A\cup B$ .

Answer to Problem 1E

  $A\cup B=\left\{0,1,2,3,4,5,6,7,8,9\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A\cup B=\left\{x:x\in A\text{or}x\in B\right\}$

Calculation:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\}\\ \Rightarrow A\cup B=\left\{0,1,2,3,4,5,6,7,8,9\right\}\end{array}$

b.

To determine

To find $A\cap B$ .

Answer to Problem 1E

  $A\cap B=\varphi$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A\cap B=\left\{x:x\in A\text{and}x\in B\right\}$

Calculation:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\}\\ \Rightarrow A\cup B=\varphi \end{array}$

Because there is no common elements in the sets A and B.

c.

To determine

To find $A-B$ .

Answer to Problem 1E

  $A-B=\left\{1,3,5,7,9\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A-B=\left\{x:x\in A\text{and}x\notin B\right\}$

Calculation:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\}\\ \Rightarrow A-B=\left\{1,3,5,7,9\right\}\end{array}$

d.

To determine

To find $A-\left(B-C\right)$ .

Answer to Problem 1E

  $A-\left(B-C\right)=\left\{1,3,5,7,9\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A-B=\left\{x:x\in A\text{and}x\notin B\right\}$

Calculation:

  $\begin{array}{l}B-C=\left\{0,6\right\}\\ \therefore A-\left(B-C\right)=\left\{1,3,5,7,9\right\}\end{array}$

e.

To determine

To find $\left(A-B\right)-C$ .

Answer to Problem 1E

  $\left(A-B\right)-C=\varphi$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A-B=\left\{x:x\in A\text{and}x\notin B\right\}$

Calculation:

  $\begin{array}{l}A-B=\varphi \\ \therefore \left(A-B\right)-C=\varphi \end{array}$

f.

To determine

To find $A\cup \left(C\cap D\right)$ .

Answer to Problem 1E

  $A\cup \left(C\cap D\right)=\left\{1,2,3,5,7,8,9\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $\begin{array}{l}A\cup B=\left\{x:x\in A\text{or}x\in B\right\}\\ A\cap B=\left\{x:x\in A\text{and}x\in B\right\}\end{array}$

Calculation:

  $\begin{array}{l}C\cap D=\left\{1,2,5,7,8\right\}\\ \therefore A\cup \left(C\cap D\right)=\left\{1,2,3,5,7,8,9\right\}\end{array}$

g.

To determine

To find $\left(A\cap C\right)\cap D$ .

Answer to Problem 1E

  $\left(A\cap C\right)\cap D=\left\{1,5,7\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $A\cap B=\left\{x:x\in A\text{and}x\in B\right\}$

Calculation:

  $\begin{array}{l}A\cap C=\left\{1,5,7\right\}\\ \therefore \left(A\cap C\right)\cap D=\left\{1,5,7\right\}\end{array}$

h.

To determine

To find $A\cap \left(B\cup C\right)$ .

Answer to Problem 1E

  $A\cap \left(B\cup C\right)=\left\{1,5,7\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $\begin{array}{l}A\cup B=\left\{x:x\in A\text{or}x\in B\right\}\\ A\cap B=\left\{x:x\in A\text{and}x\in B\right\}\end{array}$

Calculation:

  $\begin{array}{l}B\cup C=\left\{0,1,2,4,5,6,7,8\right\}\\ \therefore A\cap \left(B\cup C\right)=\left\{1,5,7\right\}\end{array}$

i.

To determine

To find $\left(A\cap B\right)\cup \left(A\cap C\right)$ .

Answer to Problem 1E

  $\left(A\cap B\right)\cup \left(A\cap C\right)=\left\{1,5,7\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $\begin{array}{l}A\cup B=\left\{x:x\in A\text{or}x\in B\right\}\\ A\cap B=\left\{x:x\in A\text{and}x\in B\right\}\end{array}$

Calculation:

  $\begin{array}{l}A\cap B=\varphi \\ A\cap C=\left\{1,5,7\right\}\\ \therefore \left(A\cap B\right)\cup \left(A\cap C\right)=\left\{1,5,7\right\}\end{array}$

j.

To determine

To find $\left(A\cup B\right)-\left(C\cap D\right)$ .

Answer to Problem 1E

  $\left(A\cup B\right)-\left(C\cap D\right)=\left\{0,3,6,9\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}A=\left\{1,3,5,7,9\right\},B=\left\{0,2,4,6,8\right\},C=\left\{1,2,4,5,7,8\right\}\\ D=\left\{1,2,3,5,6,7,8,9,10\right\}\end{array}$

Concept Used:

  $\begin{array}{l}A\cup B=\left\{x:x\in A\text{or}x\in B\right\}\\ A\cap B=\left\{x:x\in A\text{and}x\in B\right\}\\ A-B=\left\{x:x\in A\text{and}x\notin B\right\}\end{array}$

Calculation:

  $\begin{array}{l}A\cup B=\left\{0,1,2,3,5,6,7,8,9\right\}\\ C\cap D=\left\{1,2,5,7,8\right\}\\ \therefore \left(A\cup B\right)-\left(C\cap D\right)=\left\{0,3,6,9\right\}\end{array}$