= {odd numbers between 0 and 10}
= {even numbers between 0 and 10}
RESULTS BOX:
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} = {hearts, diamonds, clubs, spades} = {jacks, queens, kings, aces} RESULTS BOX: |
| |
= {2, 4, 6, 8, 10} = {6, 9} ∩ RESULTS BOX: |
| |
= {2, 3, 5, 6, 7} = {2, 3, 5, 7, 11} = {2, 3, 5, 7, 11, 15} RESULTS BOX: |
| |
∪ ∩ = RESULTS BOX: |
Venn diagrams are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way. Venn diagram, introduced by John Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to denote the relationship between sets. A Venn diagram is also called a set diagram or a logic diagram showing different set operations such as the intersection of sets, union of sets and difference of sets. It is also used to depict subsets of a set.
For example, a set of natural numbers is a subset of whole numbers, which is a subset of integers. The relation between the sets of natural numbers, whole numbers and integers can be shown by the Venn diagram, where the set of integers is the universal set . See the figure below.
Here, W represents whole numbers and N represents natural numbers
The universal set (U) is usually represented by a closed rectangle, consisting of all the sets. The sets and subsets are shown by using circles or oval shapes.
A diagram used to represent all possible relations of different sets. A Venn diagram can be represented by any closed figure, whether it be a Circle or a Polygon (square, hexagon, etc.). But usually, we use circles to represent each set.
In the above figure, we can see a Venn diagram, represented by a rectangular shape about the universal set, which has two independent sets, X and Y. Therefore, X and Y are disjoint sets. The two sets, X and Y, are represented in a circular shape. This diagram shows that set X and set Y have no relation between each other, but they are a part of a universal set.
For example, set X = {Set of even numbers} and set Y = {Set of odd numbers} and Universal set, U = {set of natural numbers}
We can use the below formula to solve the problems based on two sets.
n(X ⋃ Y) = n(X) + n(Y) – n(X ⋂ Y)
The formula used to solve the problems on Venn diagrams with three sets is given below:
n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) – n(A ⋂ B) – n(B ⋂ C) – n(A ⋂ C) + n(A ⋂ B ⋂ C)
The symbols used while representing the operations of sets are:
To draw a Venn diagram, first, the universal set should be known. Now, every set is the subset of the universal set (U). This means that every other set will be inside the rectangle which represents the universal set.
So, any set A (shaded region) will be represented as follows:
Where U is a universal set.
We can say from fig. 1 that
All the elements of set A are inside the circle. Also, they are part of the big rectangle which makes them the elements of set U.
In set theory, there are many operations performed on sets, such as:
etc. The representations of different operations on a set are as follows:
A’ is the complement of set A (represented by the shaded region in fig. 2). This set contains all the elements which are not there in set A.
It is clear that from the above figure,
A + A’ = U
It means that the set formed with elements of set A and set A’ combined is equal to U.
The complement of a complement set is a set itself.
Properties of Complement of set:
A intersection B is given by: A ∩ B = {x : x ∈ A and x ∈ B}.
This represents the common elements between set A and B (represented by the shaded region in fig. 3).
Intersection of two Sets
Properties of the intersection of sets operation:
A union B is given by: A ∪ B = {x | x ∈A or x ∈B} .
This represents the combined elements of set A and B (represented by the shaded region in fig. 4).
Union of two sets
Some properties of Union operation:
(A ∪ B)’ : This is read as complement of A union B . This represents elements which are neither in set A nor in set B (represented by the shaded region in fig. 5).
Complement of A U B
(A ∩ B)’: This is read as complement of A intersection B . This represents elements of the universal set which are not common between set A and B (represented by the shaded region in fig. 6).
Complement of A ∩ B
A – B : This is read as A difference B . Sometimes, it is also referred to as ‘ relative complement ’. This represents elements of set A which are not there in set B(represented by the shaded region in fig. 7).
Difference between Two Sets
A ⊝ B: This is read as a symmetric difference of set A and B . This is a set which contains the elements which are either in set A or in set B but not in both (represented by the shaded region in fig. 8).
Symmetric difference between two sets
Example: In a class of 50 students, 10 take Guitar lessons and 20 take singing classes, and 4 take both. Find the number of students who don’t take either Guitar or singing lessons.
Let A = no. of students who take guitar lessons = 10.
Let B = no. of students who take singing lessons = 20.
Let C = no. of students who take both = 4.
Now we subtract the value of C from both A and B. Let the new values be stored in D and E.
D = 10 – 4 = 6
E = 20 – 4 = 16
Now logic dictates that if we add the values of C, D, E and the unknown quantity “X”, we should get a total of 50 right? That’s correct.
So the final answer is X = 50 – C – D – E
X = 50 – 4 – 6 – 16
Venn’s diagrams are particularly helpful in solving word problems on number operations that involve counting. Once it is drawn for a given problem, the rest should be a piece of cake.
What do you mean by venn diagram, how do venn diagrams work, how do you do venn diagrams in math, what does a ∩ b mean, what are the different types of venn diagrams, what are the four benefits of using venn diagrams.
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Venn diagram, also known as Euler-Venn diagram is a simple representation of sets by diagrams. The usual depiction makes use of a rectangle as the universal set and circles for the sets under consideration.
In CAT and other MBA entrance exams, questions asked from this topic involve 2 or 3 variable only. Therefore, in this article we are going to discuss problems related to 2 and 3 variables.
Let's take a look at some basic formulas for Venn diagrams of two and three elements.
n ( A ∪ B) = n(A ) + n ( B ) - n ( A∩ B) n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) - n ( A ∩ B) - n ( B ∩ C) - n ( C ∩ A) + n (A ∩ B ∩ C )
And so on, where n( A) = number of elements in set A. Once you understand the concept of Venn diagram with the help of diagrams, you don’t have to memorize these formulas.
Where; X = number of elements that belong to set A only Y = number of elements that belong to set B only Z = number of elements that belong to set A and B both (AB) W = number of elements that belong to none of the sets A or B From the above figure, it is clear that n(A) = x + z ; n (B) = y + z ; n(A ∩ B) = z; n ( A ∪ B) = x +y+ z. Total number of elements = x + y + z + w
Where, W = number of elements that belong to none of the sets A, B or C
Tip: Always start filling values in the Venn diagram from the innermost value.
Example 1: In a college, 200 students are randomly selected. 140 like tea, 120 like coffee and 80 like both tea and coffee.
Solution: The given information may be represented by the following Venn diagram, where T = tea and C = coffee.
Example 2: In a survey of 500 students of a college, it was found that 49% liked watching football, 53% liked watching hockey and 62% liked watching basketball. Also, 27% liked watching football and hockey both, 29% liked watching basketball and hockey both and 28% liked watching football and basket ball both. 5% liked watching none of these games.
Solution: n(F) = percentage of students who like watching football = 49% n(H) = percentage of students who like watching hockey = 53% n(B)= percentage of students who like watching basketball = 62% n ( F ∩ H) = 27% ; n (B ∩ H) = 29% ; n(F ∩ B) = 28% Since 5% like watching none of the given games so, n (F ∪ H ∪ B) = 95%. Now applying the basic formula, 95% = 49% + 53% + 62% -27% - 29% - 28% + n (F ∩ H ∩ B) Solving, you get n (F ∩ H ∩ B) = 15%.
Now, make the Venn diagram as per the information given. Note: All values in the Venn diagram are in percentage.
To know the importance of this topic, check out some previous year CAT questions from this topic:
Solution: It is given that 200 candidates scored above 90th percentile overall in CET. Let the following Venn diagram represent the number of persons who scored above 80 percentile in CET in each of the three sections:
2. From the given condition, g is a multiple of 5. Hence, g = 20. The number of candidates at or above 90th percentile overall and at or above 80th percentile in both P and M = e + g = 60.
3. In this case, g = 20. Number of candidates shortlisted for AET = d + e + f + g = 10 + 40 + 100 + 20 = 170
4. From the given condition, the number of candidates at or above 90th percentile overall and at or above 80th percentile in P in CET = 104. The number of candidates who have to sit for separate test = 296 + 3 = 299.
Another type of questions asked from this topic is based on maxima and minima. We have discussed this type in the other article.
You can also post in the comment section below, any query or explanation for any concept mentioned in the article.
Permutation and combination and probability for cat.
Updated on: 13 September 2022
Venn diagrams define all the possible relationships between collections of sets. The most basic Venn diagrams simply consist of multiple circular boundaries describing the range of sets.
The overlapping areas between the two boundaries describe the elements which are common between the two, while the areas that aren’t overlapping house the elements that are different. Venn diagrams are used often in math that people tend to assume they are used only to solve math problems. But as the 3 circle Venn diagram below shows it can be used to solve many other problems.
Though the above diagram may look complicated, it is actually very easy to understand. Although Venn diagrams can look complex when solving business processes understanding of the meaning of the boundaries and what they stand for can simplify the process to a great extent. Let us have a look at a few examples which demonstrate how Venn diagrams can make problem solving much easier.
The first Venn diagram example demonstrates a company’s employee shortlisting process. The Human Resources department looks for several factors when short-listing candidates for a position, such as experience, professional skills and leadership competence. Now, all of these qualities are different from each other, and may or may not be present in some candidates. However, the best candidates would be those that would have all of these qualities combined.
The candidate who has all three qualities is the perfect match for your organization. So by using simple Venn Diagrams like the one above, a company can easily demonstrate its hiring processes and make the selection process much easier.
A colorful and precise Venn diagram like the above can be easily created using our Venn diagram software and we have professionally designed Venn diagram templates for you to get started fast too.
The second Venn diagram example takes things a step further and takes a look at how a company can use a Venn diagram to decide a suitable office location. The decision will be based on economic, social and environmental factors.
In a perfect scenario you’ll find a location that has all the above factors in equal measure. But if you fail to find such a location then you can decide which factor is most important to you. Whatever the priority because you already have listed down the locations making the decision becomes easier.
The last example will reflect on how one of the life’s most complicated questions can be easily answered using a Venn diagram. Choosing a dream job is something that has stumped most college graduates, but with a single Venn diagram, this thought process can be simplified to a great extent.
First, single out the factors which matter in choosing a dream job, such as things that you love to do, things you’re good at, and finally, earning potential. Though most of us dream of being a celebrity and coming on TV, not everyone is gifted with acting skills, and that career path may not be the most viable. Instead, choosing something that you are good at, that you love to do along with something that has a good earning potential would be the most practical choice.
A job which includes all of these three criteria would, therefore, be the dream job for someone. The three criteria need not necessarily be the same, and can be changed according to the individual’s requirements.
So you see, even the most complicated processes can be simplified by using these simple Venn diagrams.
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Great article, and all true, but.. I hate venn diagrams! I don’t know why, they’ve just never seemed to work for me. Frustrating!
Hey thanks for writing. It helped me in many ways Thanks again 🙂
Hi Nishadha,
Nice article! I love Venn Diagrams because nothing comes to close to expressing the logical relationships between different sets of elements that well. With Microsoft Word 2003 you can create fantastic looking and colorful Venn Diagrams on the fly, with as many elements and colors as you need.
Hi Worli, Yes, Venn diagrams are a good way to solve problems, it’s a shame that it’s sort of restricted to the mathematics subject. MS Word do provides some nice options to create Venn diagrams, although it’s not the cheapest thing around.
Please enter an answer in digits: 20 + eight =
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Venn diagrams with 3 circles: video lesson, what is the purpose of venn diagrams.
A Venn diagram is a type of graphical organizer which can be used to display similarities and differences between two or more sets. Circles are used to represent each set and any properties in common to both sets will be written in the overlap of the circles. Any property unique to a particular set is written in that circle alone.
For example, here is a Venn diagram comparing and contrasting dogs and cats.
The Venn diagram shows the following information:
Both dogs and cats:
A Venn diagram with three circles is called a triple Venn diagram.
A Venn diagram with three circles is used to compare and contract three categories. Each circle represents a different category with the overlapping regions used to represent properties that are shared between the three categories.
For example, a triple Venn diagram with 3 circles is used to compare dogs, cats and birds.
Dogs, cats and birds can all have claws and can also be pets.
Only birds:
Only both dogs and cats:
Only both dogs and birds:
Only both cats and birds:
Make a Venn Diagram for the following situation:
30 students were asked which sports they play.
When making a Venn diagram, it is important to complete any overlapping regions first.
In this example, we start with the students that play all three sports. 7 students play all three sports.
The number 7 is placed in the overlap of all 3 circles. The shaded region shown is the overlapping area of all three circles.
2. Write the remaining number of items belonging each pair of the sets in their overlapping regions
There are 3 regions in which exactly two circles overlap.
There is the overlap of basketball and tennis, basketball and football and then tennis and football.
There are 10 students that play both basketball and tennis. The overlapping region of these two circles is shown below. We already have the 7 students that play all three sports in this region.
Therefore we only need 3 more students who play basketball and tennis but do not play football to make the total of this region add up to 10.
The next overlapping region of two circles is those that play basketball and football. There are 11 students in total that play both.
The overlapping region of the basketball and football circles is shown below.
There are already 7 students who play all three sports and so, a further 4 students must play both basketball and football but not tennis in order to make the total in this shaded region add up to 11 students.
The next overlapping region of two circles is those that play football and tennis. There are 9 students in total that play both.
The overlapping region of the football and tennis circles is shown below.
There are already 7 students who play all three sports and so, a further 2 students must play both football and tennis but not basketball in order to make the total in this shaded region add up to 9 students.
Write the remaining number of items belonging to each individual set in the non-overlapping region of each circle
There are three individual sets which are represented by the three circles. There are those that play basketball, football and tennis.
20 students play basketball in total. These 20 students are shown by the shaded circle below.
We already have 3, 7 and 4 students in the overlapping regions. This is a total of 14 students so far. We need a further 6 students who only play basketball in order for the numbers in this circle to make a total of 20.
The next individual sport is football. 16 students play football in total.
There are already 4, 7 and 2 students in the overlapping regions. This makes a total of 13 students so far.
3 more students are required to make the circle total up to 16. 3 students play only football and not basketball and tennis.
Finally, there are 15 students who play tennis shown by the shaded region below.
There are already 3, 7 and 2 students in the overlapping regions, making a total of 12 students.
A further 3 students are required to make the total of 15 students in this circle.
3 students play tennis but not basketball or football.
The values in each circle sum to 28 students.
That is 6 + 4 + 3 + 7 + 3 + 2 + 3 = 28.
Since there are 30 students who were asked in total, a further 2 students must play none of these three sports.
To solve a Venn diagram with 3 circles, start by entering the number of items in common to all three sets of data. Then enter the remaining number of items in the overlapping region of each pair of sets. Enter the remaining number of items in each individual set. Finally, use any known totals to find missing numbers.
Venn diagrams are particularly useful for solving word problems in which a list of information is given about different categories. Numbers are placed in each region representing each statement.
100 people were asked which pets they have.
How many people just have a dog?
Start by entering the number of items in common to all three sets of data
3 people own all three pets and so, a number 3 is written in the overlapping region of all three circles.
Then enter the remaining number of items in the overlapping region of each pair of sets
10 people have just a dog and a rabbit.
Since 3 people are already in this region, 7 more people are needed.
21 people have just a dog and a cat.
Since 3 people are already in this region, 18 more people are needed.
7 people have just a cat and a rabbit.
Since 3 people are already in this region, 4 more people are needed.
Enter the remaining number of items in each individual set
32 people in total have a cat.
There are already 18 + 3 + 4 = 25 people in this circle.
Therefore a further 7 people are needed in this circle to make 32.
7 people just own a cat and no other pet.
18 people in total have a rabbit.
There are already 7 + 3 + 4 = 14 people in this circle.
Therefore a further 4 people are needed in this circle to make 18.
4 people just own a rabbit and no other pet.
Finally, use any known totals to find missing numbers
We are now told that 25 people own none of these pets. This means that a 25 is written outside of all of the circles but still within the Venn diagram.
The question requires the number of people who just own a dog.
There are 100 people in total and so, all of the numbers in the complete Venn diagram must add up to 100.
Adding the numbers so far, 3 + 7 + 4 + 18 + 4 + 7 + 25 = 68 people in total.
Since the numbers must add to 100, there must be a further 32 people who own a dog.
Now all of the numbers in the Venn diagram add to 100.
Here is a downloadable template for a blank Venn Diagram with 3 circles.
Here are some examples of shading Venn diagrams with 3 sets:
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We will also cover problem-solving questions. Each question is followed by a worked solution. How to solve Venn diagram questions ... students learn to use set notation with Venn diagrams and start to find probabilities using Venn diagrams. The questions below are examples of questions that students may encounter in 6th, 7th and 8th grade ...
The best way to explain how the Venn diagram works and what its formulas show is to give 2 or 3 circles Venn diagram examples and problems with solutions. Problem-solving using Venn diagram is a widely used approach in many areas such as statistics, data science, business, set theory, math, logic and etc.
How to use the venn diagram calculator: The Venn diagram calculator, helps you to solve venn diagram problems. To leave the group name or intersection label empty, simply fill in a space. To add a new line to the label, press Enter after first line and continue to the second line. When you enter a value, the Venn diagram calculator will update ...
Venn diagram questions with solutions are given here for students to practice various questions based on Venn diagrams.These questions are beneficial for both school examinations and competitive exams. Practising these questions will develop a skill to solve any problem on Venn diagrams quickly.. Venn diagrams were first introduced by John Venn to represent various propositions in a ...
Information provided in the problem usually tells you how many elements are in each set or section. Step 5: Solve the Problem. Now, you can use the diagram to answer the question. This might involve counting the number of elements in a particular set or section of the diagram, or it might involve noticing patterns or relationships between the sets.
Purplemath. Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.
Examples of Venn Diagram. Example 1: Let us take an example of a set with various types of fruits, A = {guava, orange, mango, custard apple, papaya, watermelon, cherry}. Represent these subsets using sets notation: a) Fruit with one seed b) Fruit with more than one seed.
This math tutorial video explains Venn diagram problem solving. We show you how to solve Venn diagram survey problems, and we work a few examples (both two ...
GCSE; WJEC; Venn diagrams - WJEC Solving problems using Venn diagrams. Venn diagrams are a useful tool in the world of statistics. Once you have got to grips with these, you will be able to ...
To represent this relationship using a Venn diagram, the set of plants will be our universal set and the set of trees will be the subset. Recall that this relationship is expressed symbolically as: Trees ⊂ Plants. Trees ⊂ Plants. To create a Venn diagram, first we draw a rectangle and label the universal set " U = Plants. U = Plants.
Here is an example on how to solve a Venn diagram word problem that involves three intersecting sets. Problem: 90 students went to a school carnival. 3 had a hamburger, soft drink and ice-cream. 24 had hamburgers. 5 had a hamburger and a soft drink. 33 had soft drinks. 10 had a soft drink and ice-cream. 38 had ice-cream. 8 had a hamburger and ...
These Venn Diagram Worksheets will produce three problems with a maximum of 4 questions for each Venn Diagram for the students to answer. Set Notation Problems Using Three Sets Worksheets These Venn Diagram Worksheets are great for practicing solving set notation problems of different sets, unions, intersections, and complements with three sets.
A Venn diagram is a diagram that shows the relationship between and among a finite collection of sets. If we have two or more sets, we can use a Venn diagram to show the logical relationship among these sets as well as the cardinality of those sets. In particular, Venn Diagrams are used to demonstrate De Morgan's Laws. Venn diagrams are also useful in illustrating relationships in ...
T means the set of Tennis players. V means the set of Volleyball players. The Venn Diagram is now like this: Union of 3 Sets: S ∪ T ∪ V. You can see (for example) that: drew plays Soccer, Tennis and Volleyball. jade plays Tennis and Volleyball. alex and hunter play Soccer, but don't play Tennis or Volleyball. no-one plays only Tennis.
This video solves two problems using Venn Diagrams. One with two sets and one with three sets.Complete Video List at http://www.mathispower4u.com
Example 1: Given set R is the set of counting numbers less than 7. Draw and label a Venn diagram to represent set R and indicate all elements in the set. Analysis: Draw a circle or oval. Label it R. Put the elements in R. Solution: Notation: R = {counting numbers < 7} Example 2: Given set G is the set of primary colors.
Often questions require you to draw a Venn diagram from the information given and calculate one of the value that is missing. E.g. 1 In a group of 24 girls, ... Worksheet Solving problems using Venn diagrams Qu 1-3 9-1 class textbook: p246 M8.7 Qu 1-6 (Look at Qu 2 and 5 in class) A*-G class textbook: No exercise
Venn's diagrams are particularly helpful in solving word problems on number operations that involve counting. Once it is drawn for a given problem, the rest should be a piece of cake. Venn Diagram Questions. Out of 120 students in a school, 5% can play Cricket, Chess and Carroms.
It is important to carefully list the conditions given in the question in the form of a Venn diagram. While solving such questions, avoid taking many variables. Try solving the questions using the Venn diagram approach and not with the help of formulae.
A simple Venn diagram example. The overlapping areas between the two boundaries describe the elements which are common between the two, while the areas that aren't overlapping house the elements that are different. Venn diagrams are used often in math that people tend to assume they are used only to solve math problems.
2.2: Venn Diagrams. Page ID. Julie Harland. MiraCosta College. This is a Venn diagram using only one set, A. This is a Venn diagram Below using two sets, A and B. This is a Venn diagram using sets A, B and C. Study the Venn diagrams on this and the following pages. It takes a whole lot of practice to shade or identify regions of Venn diagrams.
To solve a Venn diagram with 3 circles, start by entering the number of items in common to all three sets of data. Then enter the remaining number of items in the overlapping region of each pair of sets. Enter the remaining number of items in each individual set. Finally, use any known totals to find missing numbers.
Recommendations. Skill plans. IXL plans. Virginia state standards. Textbooks. Test prep. Improve your math knowledge with free questions in "Use Venn diagrams to solve problems" and thousands of other math skills.