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Chapter 2: Problem 57

When filling in cardinalities for regions in a two-set Venn diagram, the innermost region, the intersection of the two sets, should be the last region to be filled in.

Short Answer

Expert verified
To fill in a two-set Venn diagram, first fill in the regions unique to each set, then fill in the intersection last to avoid counting elements more than once that exist in both sets. The intersection is filled last to evaluate any commonalities.

Step by step solution

01

Understand the scenario and the sets

Identify the two sets of items or participants that you want to compare or analyze. This will form the basis of your Venn diagram. This is usually part of the question you're being asked.
02

Draw the Venn Diagram

First, represent these sets graphically. Draw two circles that overlap. Each circle represents a set. The space where the circles overlap represents the intersection of the two sets 鈥� that is, the items that are common to both sets.
03

Fill in the Regions

Start by filling in the regions of the Venn Diagram that do not intersect - these represent the parts of our sets that are unique to each. We want to fill in the intersection last, as the exercise states, to ensure we're not duplicating any counts.
04

Fill in the Intersection

After filling in all other regions, the region that represents the intersection of the two sets should be filled in. This intersection represents elements that are common in both sets.

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Most popular questions from this chapter

In Exercises 29-32, use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \(n(U)=38, n(A)=26, n(B)=21, n(C)=18\) \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=8\) \(n(A \cap B \cap C)=7\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(B \cap C\)

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A \cap(B \cup C)\)

A survey of 180 college men was taken to determine participation in various campus activities. Forty-three students were in fraternities, 52 participated in campus sports, and 35 participated in various campus tutorial programs. Thirteen students participated in fraternities and sports, 14 in sports and tutorial programs, and 12 in fraternities and tutorial programs. Five students participated in all three activities. Of those surveyed, a. How many participated in only campus sports? b. How many participated in fraternities and sports, but not tutorial programs? c. How many participated in fraternities or sports, but not tutorial programs? d. How many participated in exactly one of these activities?

Assume \(A \neq B\). Draw a Venn diagram that correctly illustrates the relationship between the sets. \(A \cap B=B\)
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