So far in this unit, we've talked about sets and Venn Diagrams, as well as everything that goes into them. In the previous lesson, we learned about creating Venn Diagrams with two sets or two circles. In this section, we are going to be working with Venn Diagrams involving three sets or three circles. It is important to note that Venn Diagrams do not have a limit to the number of sets you can use. Once you move beyond three sets, the complexity of the Venn diagram increases significantly.
Three-circle Venn Diagrams are useful when trying to make conclusions about different sets. It holds the same ideas and principals that two-circle Venn Diagrams have. Let's look at what a three-circle Venn diagram tells us in the figure below.
We can see the similarity between this and the two-circle Venn diagram found in the previous section. From looking at the three-circle Venn diagram, we can conclude that each set contains the following sections/regions:
Now let's look at what each region tells us specifically about each set:
Just like with two-circle Venn diagrams, it is important to know what each section/region represents in order to help us solve problems and/or identify sets.
A survey was conducted about radio listening habits of adults in the ages of 18 to 45. Those surveyed were asked to identify which genre of music they listen to: Hip-Hop, Country, or Rock-and-Roll. Construct a generic Venn diagram that would represent the results of this survey, then identify the set of people and the sections represented by $H\cap (C\cap R)'$ where:
H = The set of all people who listen to Hip-Hop
C = The set of all people who listen to Country
R = The set of all people who listen to Rock-and-Roll
This is where we are going to employ the process that we learned in the previous section to help us.
$$C\cap R = \{IV,V,VI,VII\}\cap \{II,III,IV,VII\}=\{IV,VII\}$$
Next, we find the compliment of that, so everything that is in Set U but not in that set.
$$(C\cap R)'=\{I,II,III,IV,V,VI,VII,VIII\}-\{IV,VII\}=\{I,II,III,V,VI,VIII\}$$
Lastly, intersect that with Set H:
$$H\cap (C\cap R)'=\{I,II,VI,VII\}\cap \{I,II,III,V,VI,VIII\}=\{I,II,VI\}$$
We can see from the Venn diagram that this represents the set of all people who like Hip-Hop music, Hip-Hop and Rock-and-Roll, and Hip-Hop and Country, but do not like all three.
Sometimes we need to solve problems involving three-circle Venn diagrams. To do that, we will employ the same process that we did with two-circle Venn diagrams. As we increase in difficulty with each problem, it becomes even more important that we understand what each region of the circle represents.
Fifty students were surveyed and asked if they were taking a Social Science (SS), Humanities (HM), or a Natural Science (NS) course in their next semester. It was found that 21 students were taking a Social Science course, 19 were taking a Natural Science course, and 26 were taking a Humanities course. Additionally, it was found that 7 were taking a Social Science and Natural Science course, 9 were taking a Social Science and Humanities course, and 10 were taking a Humanities and Natural Science course. Further results also showed that 3 students were taking all three courses, and that 7 students were taking none of the three courses in the next semester. How many students are only taking a Social Science course?
To start finding the solution, let's first look at the Venn diagram and the regions:
$$\text{SS} =\{I,II,VI,VII\}$$
$$\text{HM} =\{II,III,IV,VII\}$$
$$\text{NS} =\{IV,V,VI,VII\}$$
Now let's pull the numbers from the problem and see if we can arrive at a conclusion. We were told the following:
| Cardinality | Section Numbers | Cardinality | Section Numbers |
|---|---|---|---|
| n(SS)=21 | {I,II,VI,VII} | n(HM)=26 | {II,III,IV,VII} |
| n(NS)=19 | {IV,V,VI,VII} | n(SS ∩ NS)=7 | {VI,VII} |
| n(SS ∩ HM)=9 | {II,VII} | n(HM ∩ NS)=10 | {IV,VII} |
| n(SS ∩ NS ∩ HM)=3 | {VII} | n(none)=7 | {VIII} |
When we look at all the information like that, it really starts to help us paint a picture. We can see that most of the cardinalities are represented by multiple regions except two. To start figuring out the number of elements (cardinality) in each region, it will be easiest to start with the innermost region of our diagram (VII) and start to spiral our way out.
Since we see that 3 students are in section VII, we can move out of the innermost part and into the intersections of two of the circles. In this case, we will move to the intersection of Social Science and Humanities, which tells us there are 9 total students in the regions II and VII. If we know there are 3 students in region VII, that means region II has $9-3 =6$ students. Next, we would move onto another intersection, Humanities and Natural Science. They have 10 between both sections, meaning that section IV would contain $10-3 =7$. Lastly, move to the last intersection to find that region VI would contain $7-3=4$. Before we move on to the individual regions of I, III, and V, let's look at our Venn diagram:
Note that in the above Venn diagram, we replaced the section numbers with the cardinality; we just did this for the sake of space. The regions are still labeled accordingly. Next, we are going to find the remaining people who would be in regions I, III, and V.
The number of elements in region I $=21-6-3-4=8$.
The number of elements in region III $=26-6-3-7=10$.
The number of elements in region V $=19-7-4-3=5$.
Finally producing us with our finished Venn diagram:
Now we can answer the question of "How many students are only taking a Social Science course?" Based on our Venn diagram, that would be the set of students who fell only in region I, which is a total of 8 students.
One hundred-fifty people were surveyed and asked if they believed in UFOs, Ghosts, and Bigfoot. The results are given below:
| 43 believed in UFOs | 44 believed in ghosts |
| 25 believed in Bigfoot | 10 believed in UFOs and ghosts |
| 8 believed in ghosts and Bigfoot | 5 believed in UFOs and Bigfoot |
| 2 believed in all three |
How many people surveyed believed in at least one of these things?
To find the solution to this problem, we will approach it very similar to the previous example. This time, we aren't going to break it down as far. Since this problem models the same idea as the previous problem, we will start at the very center of our Venn diagram with the intersection of all three circles (region VII), and then spiral our way out.
Region VII: This would be the two who believed in all three.
Region II: This would be UFOs and ghosts plus the center, thus $10-2=8$
Region IV: This would be ghosts and Bigfoot plus the center, thus $8-2=6$
Region VI: This would be UFOs and Bigfoot plus the center, thus $5-2=3$
Now we can start to find how many in the remaining areas.
Region I: UFOs $43-8-2-3=30$
Region III: Ghosts $44-8-6-2=28$
Region V: Bigfoot $25-3-2-6=14$
Lastly, we would need to find out how many are in region VIII $150-30-8-28-6-14-3-2=59$. This gives us our final Venn diagram of:
From this, we can conclude that $150-59 =91$ people believed in at least one of these three things.
A researcher was interested in finding three ways to live a healthier lifestyle. In a study of 690 people, it was found that the following were the best ways to improve health: Sixty-two said diet only, 36 said exercise only, and 93 said sleep habits only. There were 370 total people who suggested diet as a factor, 159 said that sleep habits and diet but not exercise was a factor, and 23 said that exercises and diet but not sleep habits was a factor. Lastly, 585 people said that at least one of these three factors impact a healthier lifestyle. Draw the Venn diagram and answer the following questions.
First we need to start pulling out the important information in this question and putting it into some notation that we understand.
Unlike the other examples, we were not given the innermost part region VII. Instead, we were given other pieces and from those we have to start and build this puzzle. First, start by entering the single sections we do have:
Next, look to see what is left in the list created and see what can be determined from that. Looking at the list, it can be seen at bullet 5 that $n(I,II,VI,VII)=370$. Looking at our diagram, it can be seen that we have all but VII. Thus, if we subtract all the other sections from 370, we'd end up with the number for section VII, $370-62-23-159=126$. This means there are 126 people who felt that diet, exercise, and sleep were important factors of a healthier lifestyle. Placing that number in the correction section of the Venn diagram shows us that the only section we have left to find inside the circles is IV. This can be found by using the information in our last bullet $585-62-23-36-93-159-126=86$. To finish out this diagram, we need to identify how many people didn't feel that any of these are factors of a healthier lifestyle. To do that, we would use the information from our last bullet to get $690-585=105$ people. This provides us with the diagram of:
Now we are ready to answer the questions.
$n(\text{Diet}\cap\text{Exercise}\cap\text{Sleep})=126$
There are three sections that are exactly two of the three factors. This would be regions II, IV, and VI, meaning our answer is $23+86+159=268$.
Lastly, the number of people who felt exercise was factor is $23+36+86+126=271$. To find the percentage, we take the part and divide it by the whole to get $271/690 = 0.39$ and then multiply it by $100\%$, which would give us $39\%$.
When it comes to solving problems involving Venn Diagrams, it is extremely important to pick up on the words that signify each individual region and what they represent in terms of the entire operation. If the problem had been talking about "only", then we would be interested in just a single section. When we talk about generalizations, it broadens the number of sections/regions we are looking at.