Chapter 2.3, Problem 2TTO

To determine

To sketch: A Venn diagram and shade the section $A^{\prime }\cup B$.

Answer to Problem 2TTO

A required Venn diagram as shown below.

Explanation of Solution

Step 1:

Draw a Venn diagram and name each area with a Roman numeral.

Step 2:

From the Venn diagram, list the regions that make up each set.

$U=\left\{\text{I, II, III, IV}\right\}$, $A=\left\{\text{I, II}\right\}$ and $B=\left\{\text{II, III}\right\}$.

Step 3:

Compute $A^{\prime }\cup B$ by using the sets in step 2.

The regions I and II are in $A$.

Therefore, the regions III and IV are not in A.

That is, the complement of A is III and IV.

That is, $A^{\prime }=\left\{\text{III},\text{ IV}\right\}$.

The region II, III and IV are in either $A^{\prime }$ or B, so these regions are in $A^{\prime }\cup B$.

That is, $A^{\prime }\cup B=\left\{\text{II, III, IV}\right\}$

Step 4:

By step 3, shade the region II, III and IV as shown in below Figure 2.

From the Figure 2, it is observed that shaded regions represents $A^{\prime }\cup B$.