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Addition and subtraction Types of data Representing data Frequency table Frequency graphHere you will learn about Venn diagrams, including what they are, what they represent, their uses and key set notation.
Students will first learn about Venn diagrams as part of statistics and probability in high school.
A Venn diagram is a data visualization of two or more sets.
For example,
Above are examples of a two set and a three set Venn diagram with the following features:
To analyze data using Venn diagrams, all of the values within each set must be correctly placed into the correct part of the Venn diagram. The number of sets is usually outlined or deduced from the information provided.
Use this quiz to check your students’ understanding of using Venn diagrams. 10+ questions with answers covering Venn diagrams topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your students’ understanding of using Venn diagrams. 10+ questions with answers covering Venn diagrams topics to identify areas of strength and support!
DOWNLOAD FREETo construct a Venn diagram, you draw a region containing two or more overlapping circles (or ellipses), each representing a set. Then fill in the relevant information into each part of the Venn diagram.
For example, this Venn diagram shows the set of numbers \xi={1, 2, 3, 4, 5, 6, 7, 8, 9, 10} which have been sorted into factors of 10 (F) and even numbers (E):
To describe a subset, you need to understand key symbols and set notation for different sets including the intersection of sets, the union of sets and the absolute complement of sets.
Here are some common examples of set notation and their meaning:
You can use combinations of these, for example
A^{\prime} \cup B^{\prime} or (A \cap B)^{\prime}
To calculate the number of items in a subset of a Venn diagram, add together the frequencies of the required subset.
For example, this Venn diagram shows the number of people who own a cat (C) or a dog (D).
To find the total number of people who own a dog, you need to include everyone in set D. The number of people who own a dog is 12+6=18.
To find the total number of people who own a dog or a cat, you need to include everyone in set C and set D. The number of people who own a dog or a cat is 12+6+9=27.
To calculate a Venn diagram probability, you need to know each frequency within a subset and the frequency of the larger set which the probability is conditioned on.
For example, this Venn diagram shows the number of people who like Chinese food (C) and Indian food (I):
Find the probability that a person likes Chinese and Indian food.
In general, probability =\cfrac{\text{number of desired outcomes}}{\text{total number of outcomes}}
Here, the probability that a person chosen at random likes Chinese and Indian food, written P(C\cap{I}), is \cfrac{34}{80} since 34 people like Chinese and Indian food out of 80 people in total.
How does this relate to high school math?
Each example below provides a step-by-step guide on how to solve specific types of problems involving Venn diagrams.
The set \xi={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
Draw a Venn diagram that shows the distribution of the values in the set \xi into the two following sets:
2Draw the overlapping circles within the universe and label.
The first circle should be labeled with the name of the set Prime Numbers. The second circle should be labeled Factors of 12.
3Fill in the information for each subset within the Venn diagram.
Place each item in the universal set into the correct part of the Venn diagram.
Top tip: Go through each item in order so you don’t miss any values.
So far you have the following Venn diagram:
Continuing this process, you get the final solution:
A research group divided 40 construction workers into three groups based on the following criteria:
10\% of the participants satisfy all three criteria.
\cfrac{1}{2} of the construction workers satisfy at least 2 of the three criteria.
Complete the following three circle Venn diagram to represent the distribution of the participants in the research study.
Draw the rectangular universe.
The universal set is already given, so move on to the next step.
Draw the overlapping circles within the universe and label.
The three sets have already been given, so move on to the next step.
Fill in the information for each subset within the Venn diagram.
Since 10\% of participants are within all three sets, the very center of the Venn diagram where all three sets overlap would contain the value:
(40\div{100})\times{10}=4
Since \cfrac{1}{2} of the participants are in at least 2 of the three sets, 40\times\cfrac{1}{2}=20. The missing value in the intersection between A and C is therefore 20-(2+4+5)=9.
The final missing value for set B is the number of remaining participants. This is equal to 40-(8+3+2+9+4+5+2)=7.
This gives us the completed Venn diagram.
State the name of the subset that is the shaded region in the Venn diagram below:
Determine which sets are included in the shaded region.
Every value in set A and every value in set B are included in the shaded region.
Determine which sets are not included in the shaded region.
Every value not in the set A or B is not included in the shaded region.
State the set notation for the shaded region.
As every region within set A or set B is shaded, this is the union of set A and set B, written as A\cup{B}.
Solution: A\cup{B}
Note: A real world example of this may be the set of students who study English or German, so A\cup{B} would represent all students that study English, German or both. A student that studied Spanish is not represented by A\cup{B}.
A meteorologist recorded the daily weather patterns over the month of November. The two factors that he wants to look at are:
Below is a Venn diagram of her results.
Determine the frequency of the set S\cup{T}^{\prime}.
Calculate the frequencies in each subset of the Venn diagram.
The subset S\cup{T}^{\prime} is the union of all of the values in the set S or all of the values not in the set T. This is the following shaded region:
Add all of the frequencies within the two sets.
Do not count any duplicated values twice. Only add each value within the two sets once to get 8+3+2=13.
Solution: S\cup{T}^{\prime}=13
40 musicians were asked if they could play the Violin (V) or the Piano (P). The results are shown in the Venn diagram below.
Calculate the probability of selecting a musician at random that can only play the violin.
Determine the parts of the Venn diagram that are in the subset.
The subset of musicians that can only play the violin is the set:
Calculate the frequency of the subset.
The frequency of musicians who can only play the violin is 16.
Calculate the total frequency of the larger set.
The larger set is every musician. Calculate this by adding all the sections together, 12 + 16 + 3 + 9 = 40.
Write the probability as a fraction, and simplify.
The probability of picking a musician at random that can only play the violin is \cfrac{16}{40}=\cfrac{2}{5}.
70 students were asked about whether they play B= {basketball} or S= {soccer}. The results are shown in the Venn diagram below.
Calculate the probability of a student not playing soccer given that they play basketball.
Determine the parts of the Venn diagram that are in the subset.
Identify the subset which is the part of B which is not a part of S:
Calculate the frequency of the subset.
The frequency of values in the subset is 18.
Calculate the total frequency of the larger set.
In this case, the student plays basketball, so the population you are looking at is those in set B and we can ignore those not in set B.
The frequency of values in set B is 18+15 = 33.
Write the probability as a fraction and simplify.
1. Let x be an integer. Construct a two set Venn diagram to represent the set \xi=\{0<x\leq{20}\} shared between the set of even numbers and the set of the multiples of 3.
2. A group of people were surveyed about their phobias.
Which Venn diagram shows the correct results of the survey?
3. Determine the shaded region on the Venn diagram for the set A^{\prime}\cap{B}^{\prime}.
The set A^{\prime}:
The set B^{\prime}:
And so the set A^{\prime}\cap{B}^{\prime} is the intersection of the two sets A^{\prime} and B^{\prime} or the regions that occur in both sets. This is the region:
4. A local pharmacy is carrying out some research. They would like to find out how many people have had one of the following three symptoms of flu:
Calculate the frequency of people who had a cough or a fever, but not a headache.
178 people had a cough only. 44 people had a fever only. 52 people had a cough and a fever, but not a headache. The number of people that had a couch or a fever, but not a headache is 178+52+44=274.
5. The Venn diagram below shows the names of 2D shapes that have at least one pair of parallel sides (P) or are quadrilateral (Q).
From the set \xi, determine P(P\cup{Q})^{\prime}.
The set (P\cup{Q})^{\prime} contains all of the values that are not in the set P or Q:
The three shapes that are in the shaded region are: equilateral triangle, regular pentagon, and scalene triangle.
As there are 11 shapes in the Venn diagram,
P(P\cup{Q})^{\prime}=\cfrac{3}{11}.
6. A university’s entry requirements to study a degree in Medical Science is a grade A in Math, and then a grade A in at least two of the three subjects: Chemistry, Biology, or Physics. The Venn diagram below shows the frequencies of students who applied for the course and achieved an A grade in each subject.
Calculate P(C given that B\cup{P}).
The frequencies within each shaded region for the required subsets are:
C\cap(B\cup{P}):
\begin{aligned} &=8+2+4 \\\ &=14 \end{aligned}
B\cup{P}:
\begin{aligned} &=8+11+2+9+4+7 \\\ &=41 \end{aligned}
P(C given that B\cup{P})=\cfrac{14}{41}
Yes, Venn diagrams are multi-circle diagrams and there can be as many circles as the context requires, which includes four sets or more.
The terms ‘set diagram’ and ‘logic diagram’ are also used to refer to Venn diagrams.
A branch of mathematics that deals with sets, which are collections of groups or objects.
Logician John Venn (1834-1923) was also a mathematician and philosopher who introduced the Venn diagram which is widely used in modern day statistics, probability, set theory and computer science.
Clarence Irving wrote about the Venn diagram in his book ‘A survey of symbolic logic’ in 1918 after which Venn diagrams became more popular.
The Venn diagram below shows the relationship between the data set of Factors of 18, and the data set of Prime numbers less than 18.
Leonhard Euler (1707-1783) was a pioneering Swiss mathematician who has greatly influenced today’s fundamental understanding of mathematics in various fields such as fluid mechanics, complex analysis and calculus.
An Euler diagram (sometimes referred to as Eulerian circles) is different to a Venn diagram because they only show relevant relationships, whereas a Venn diagram shows all possible relationships.
Take for example the set of 2D shapes and 3D shapes. These two sets cannot overlap as a 2D shape cannot be a 3D shape, however quadrilaterals are a subset of 2D shapes and so they are contained entirely within the set of 2D shapes.
A Carroll diagram (named after the author and mathematician Lewis Carroll) is similar to a two-way table which categorizes items within a set into smaller subsets.
For example, below is a Carroll diagram showing the Humans and Animals that did/not attend the tea party in the book ‘Alice in Wonderland’:
Note: The term Venn diagram is named after John Venn and so ‘Venn’ should be capitalized (this is the same for Euler diagrams and Carroll diagrams as they are named after Leonhard Euler, and Lewis Carroll respectively).
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