Until now, we have examined sets using set notation. We know from previous lessons that the following conventions are used with sets:
- Capital letters are used to denote sets.
- Lowercase letters are used to denote elements of sets.
- Curly braces { } denote a list of elements in a set.
Another way to look at sets is with a visual tool called a Venn diagram, first developed by John Venn in the 1880s. In a Venn diagram, sets are represented by shapes; usually circles or ovals. The elements of a set are labeled within the circle. Let's look at some examples.
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. Draw and label a Venn diagram to represent set G and indicate all elements in the set.
Analysis: Draw a circle or oval. Label it G. Put the elements in G.
Solution:
Notation: G = {primary colors}
Example 3: Given set B is the set of all vowels in the English alphabet. Draw and label a Venn diagram to represent set B and indicate all elements in the set.
Analysis: Draw a circle or oval. Label it B. Put the elements in B.
Solution:
Notation: B = {vowels}
In each example above, we used a Venn diagram to represent a given set pictorially. Venn diagrams are especially useful for showing relationships between sets, as we will see in the examples below. First, we will use a Venn diagram to find the intersection of two sets. The intersection of two sets is all the elements they have in common.
Example 4: Let X = {1, 2, 3} and Let Y = {3, 4, 5}. Draw and label a Venn diagram to show the intersection of sets X and Y.
Analysis: We need to find the elements that are common in both sets. Draw a picture of two overlapping circles. Elements that are common to both sets will be placed in the middle part, where the circles overlap.
Solution:
Explanation: The circle on the left represents set X and the circle on the right represents set Y. The shaded section in the middle is what they have in common. That is their intersection.
The Venn Diagram in example 4 makes it easy to see that the number 3 is common to both sets. So the intersection of X and Y is 3. This is what X and Y have in common. The intersection of X and Y is written as and is read as “X intersect Y”. So Intersection means “X and Y”. In example 5 below, we will find the union of two sets. The union of two sets is the set obtained by combining the elements of each.
Example 5: Let X = {1, 2, 3} and Let Y = {3, 4, 5}. Draw and label a Venn diagram to represent the union of these two sets.
Analysis: To find the union of two sets, we look at all the elements in the two sets together.
Solution:
Explanation: Any element in X, Y, or in their intersection is in their union. So X union Y is {1, 2, 3, 4, 5}. Both circles have been shaded to show the union of these sets.
The union of two sets is written as and is read as “X union Y”. It means “X or Y”. Let's compare intersection and union.
Intersection | Union | |
written as | ||
read as | X intersect Y | X union Y |
meaning of | X and Y | X or Y |
Look for the | elements in common to both | combine all elements |
The examples in this lesson included simple Venn diagrams. We will explore this topic in more depth in the next few lessons. We will also learn more about intersection and union in this unit.
Summary: We can use Venn diagrams to represent sets pictorially. Venn diagrams are especially useful for showing relationships between sets, such as the intersection and union of overlapping sets.
Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer, then choose a different button.
1. | Which of the following is represented by the Venn diagram below? |
2. | Which of the following is represented by the Venn diagram below? |
3. | Which of the following is represented by the Venn diagram below? |
4. | Which of the following is the correct roster notation for set X? |
5. | Which of the following relationships is shown by the Venn diagram below? |
How to Create Venn Diagram?
Want to create a Venn Diagram? We will provide you with an introduction to Venn Diagrams and let you know how you can create a Venn Diagram with our Venn Diagram software.
Venn Diagram, also called Primary Diagram, Logic Diagram or Set Diagram, is widely used in mathematics, statistics, logic, computer science and business analysis for representing the logical relationships between two or more sets of data. A Venn Diagram involves overlapping circles, which present how things are being organized, their commonality and differences.
Venn diagrams are effective in displaying the similarities or the differences between multiple data set (usually two to four). Venn diagrams also describe the result of unifying several data groups. They can also describe the overlapping responsibilities of two organizations/teams.
Listed below are some Venn diagram examples. As you can see, Venn diagrams consist of overlapped oval shapes, showing the content of the data sets as well as the data that co-exists in multiple groups.
Note that instead of ovals, Venn Diagrams can also be formed with triangles, rectangles, squares, and other shapes, although uncommon.
- Select Diagram > New from the main menu.
- In the New Diagram window, select Venn Diagram and click Next.
- Select an existing Venn Diagram template, or select Blank to create from scratch. Click Next.
- Enter the diagram name and click OK.
- Drag and drop the oval shapes from the palette onto the canvas. Double click to create a text label.
- When you finished, you can export the diagram as an image (JPG, PNG, PDF, SVG, etc) and share it with your friends or co-workers (Project > Export > Active Diagram as Image…).
What is a Venn diagram?
A Venn diagram shows the relationship between a group of different things (a set) in a visual way. Using Venn diagrams allows children to sort data into two or three circles which overlap in the middle. Each circle follows a certain rule, so any numbers or objects placed in the overlapping part (the intersection) follow both rules.
Venn diagrams in KS1
Venn diagrams encourage children to sort objects or numbers according to given criteria. Learning how to sort begins in Key Stage 1, when teachers may ask a child to sort a group of objects into two groups according to certain rules. For example, they may be given these shapes and asked to put them into either one of the following two circles:
Venn diagrams in KS2
A Venn diagram is when the two sorting circles overlap in the middle. Children need to think about how to sort something according to the two rules. For example, they might be asked to sort the numbers 5, 8, 10, 25 and 31 in the following Venn diagram:
In this example, 10 would go in the circle on the left (it's in the 5x table but not an odd number), 5 and 25 would go in the intersection (they are both part of the 5x table and odd numbers), 31 would go in the circle on the right (it's an odd number and not in the 5x table) and 8 would be outside the circles (it doesn't fit the criteria of this Venn diagram).
More advanced activities involving Venn diagrams might include reading bar charts or using their knowledge of multiples to find information to sort into a Venn diagram.Venn diagrams are a great way to combine skills: children can practise data-handling while learning about properties of shapes or number facts.
What Is A Venn Diagram: Explained For Primary Parents And Kids
In this post we will be explaining what venn diagrams are, how they can be used and what your child will be learning about them throughout primary school. We’ve also included a number of venn diagram based questions to test your child’s skills, so take a look!
This blog is part of our series of blogs designed for parents supporting home learning and looking for home learning resources during the Covid-19 epidemic.
What is a venn diagram?
A Venn diagram (named after mathematician John Venn in 1880) is a method used to sort items into groups.
Venn diagrams explained: How to interpret them
These diagrams are usually presented as two or three circles overlapping, with the overlapping sections containing items that fit into both (or all, if three circles overlap) groups. Items which don’t belong to either/any group are placed on the outside of the circles.
An example of a simple venn diagram:
In the example above, you can see that the words have been sorted into one of four categories: those with 5 letters (placed in the left circle), those with a double consonant (placed in the right circle), those with both 5 letters and a double consonant (placed in the centre overlap) and those with neither (placed round the edge).
Below is another example with three circles creating eight groups.
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Venn diagrams
Chris Joyce (2008)
A Venn diagram is a type of graphic organiser. Graphic organisers are a way of organising complex relationships visually. They allow abstract ideas to be more visible.
Although Venn diagrams are primarily a thinking tool, they can also be used for assessment. However, students must already be familiar with them before they can be used in this way.
- When to use
- Venn diagrams are used to compare and contrast groups of things.
- They are a useful tool for formative assessment because they:
- can be used to generate discussion; and
- provide teachers with information about students’ thinking.
- In science, they are helpful for classification.
- As an accepted convention for representing similarities and differences, knowing how to use them contributes to the Key Competency, Using language, symbols, and texts.
- The theory
Venn diagrams originate from a branch of mathematics called set theory. John Venn developed them in 1891 to show relationships between sets. They are now used across many other disciplines.
Information is usually presented to students in linear text. Especially when there is a lot of information, it is difficult to see relationships in this format. Venn diagrams enable students to organise information visually so they are able to see the relationships between two or three sets of items. They can then identify similarities and differences.
How the strategy works
A Venn diagram consists of overlapping circles. Each circle contains all the elements of a set. Where the circles overlap shows the elements that the set have in common. Generally there are two or three circles. Any more and the exercise becomes very complicated.
The following science example compares the features of bats and birds.
The following is a maths example:
- What to do
- Creating a Venn diagram
- If the assessment focus is on organising information:
- Students view written text, pictures, diagrams, or video/film about two (or sometimes three) items that have some related characteristics.
- Identify what items they want to compare (e.g., birds and bats).*
- Draw two overlapping circles. Label each circle (Bird, Bat).
- In each circle, fill in the characteristics of each item.
- Identify which characteristics appear in both circles. These characteristics go in the intersection (where the two circles overlap).
- Sometimes features that don't fit in either set are included. E.g., in the maths example, if all numbers between 1 and 30 were included, some would not be a multiple of either 3 or 5. These are placed outside the circles.
It is preferable that students then use their Venn diagram to compare the sets.
*Sometimes the first step is to draw a rectangle and identify the universal set. For example, in the science example above, the universal set might be Animals that fly. The circles for birds and bats are then drawn inside the rectangle.
Reading a Venn diagram
If the assessment focus is to interpret a Venn diagram:
- Ask questions about the similarities and differences that the Venn diagram illustrates.
- Provide true/false statements, e.g., 10 is a multiple of 3 and 5.
- Ask questions about, or discuss the two sets. For example, students may be able to say that bats have some similarities to birds, but are not birds because they don't lay eggs or have feathers.
- If appropriate, ask questions that encourage students to make generalisations, e.g., Can we classify a bat as a member of the bird family?
Limitations
When trialling ARB resources we have found that many students do not use Venn diagrams well. Some are unfamiliar with them. If using Venn diagrams as an assessment strategy, students must have already demonstrated that they know how they work, to ensure that the assessment is valid.
Adapting the strategy
- Venn diagrams are widely used as a tool for thinking. They are therefore also a useful teaching strategy.
- They can be useful for practising making logic statements, e.g., if/then, all/some/no, may be.
- When teaching students about Venn diagrams, or working with young students, use concrete materials, such as post-its, cards, string, or hoops, which students can move around.
- Computer programmes such as Inspiration are useful for creating Venn diagrams.
Examples of ARB resources that include Venn diagrams
What is a Venn Diagram – Explain with Examples
The term Venn diagram is not foreign since we all have had Mathematics, especially Probability and Algebra. Now, for a layman, the Venn diagram is a pictorial exhibition of all possible real relations between a collection of varying sets of items. It is made up of several overlapping circles or oval shapes, with each representing a single set or item.
Venn diagrams depict complex and theoretical relationships and ideas for a better and easier understanding. These diagrams are also professionally utilized to display complex mathematical concepts by professors, classification in science, and develop sales strategies in the business industry.
Image Source: pinterest.com
Evolution of Venn Diagram
The growth of the Venn diagram dates back to 1880 when John Venn brought them to life in an article titled ‘On the Diagrammatic and Mechanical Representation of Propositions and Reasoning.
’ It was in the Philosophical Magazine and Journal of Science. John Venn carried out a thorough research on these diagrams and foresaw their formalization. He is the one who originally generalized them, no wonder their naming, i.e.
, Venn Diagrams in 1918.
There is a small gap between Venn diagrams and Euler diagrams invented in the 18th century by Leonhard Euler, who also had a hand in its development in the 1700s. John used to refer to the charts as Eulerian circles.
The development of Venn diagrams continued in the 20thcentury. For instance, around 1963, D.
W Henderson revealed the existence of an n-Venn graph consisting of n-fold rational symmetry, which pointed out that n was a prime number.
This concept was delved into by four other intellects in the following years, who concluded that rotationally symmetric Venn diagrams only exist if n is a prime number.
Ever since, these diagrams have become part of today’s studying curriculum and illustrate business information. Venn and Euler’s diagrams got incorporated as a component of instruction in the set theory of the new math movement in the year 1960.
Why are Venn diagrams Important?
Venn diagrams are useful as a teaching and studying tools to scholars, teachers, and professors. They help represent simple mathematical concepts in grade schools as well as rocket-science kind of theories and problems among logicians and mathematicians.
Furthermore, together with Set theory, Venn diagrams have facilitated a clearer up-to-date understanding of infinite numbers and real numbers in Mathematics. They also enhanced the creation of a common language and system of symbols concerning Set theory among researchers and mathematicians.
They are ideal for illustrating similarities and differences among items or ideas when circles overlap or otherwise. This feature is commonly utilized in the business industry to find and create a niche in the market for goods and services. These facilitate incredible sales reports and huge realized profits among entrepreneurs.
You can also use Venn diagrams
Venn Diagrams
Set Not'nSets ExercisesDiag. Exercises
Venn diagrams were invented by a guy named John Venn (no kidding; that was really his name) as a way of picturing relationships between different groups of things.
Inventing this type of diagram was, apparently, pretty much all John Venn ever accomplished. To add insult to injury, much of what we refer to as “Venn diagrams” are actually “Euler” diagrams.
But we'll stick with the usual “Venn” terminology for the purposes of this lesson.
Since the mathematical term for “a group of things” is “a set”, Venn diagrams can be used to illustrate set relationships.
To draw a Venn diagram, we first draw a rectangle which is called our “universe”.
In the context of Venn diagrams, the universe is not “everything in existence”, but “everything that we're working with right now”.
Let's deal with the following list of things: moles, swans, rabid skunks, geese, worms, horses, Edmontosorum (a variety of duck-billed dinosaurs), platypodes (being more than one platypus), and a very fat cat.
(By the way, about the plural of “platypus”: It is not “platypi”. In Australian usage, the plural is often given as being “platypusses”, but the technically-correct plural is “platypode”. When platypode give birth, their babies are called “puggles”.)
- We'll call our universe “Animals”, since every element in our set is some sort of animal:
- Let's say we want to classify things according to being small and furry or being a duck-bill. We draw circles inside our universe to display our classifications:
- Now we'll fill in, or “populate”, the diagram. Moles, rabid skunks, platypusses, and my (dear departed) very fat cat are all small and furry:
- Swans, geese, platypodes, and Edmontosorum are all duck-bills:
Worms are small but not furry, and horses are furry but not small, and neither is a duck-bill. However, they are some of the animals we're considering. This means that they fit inside our universe, but they're outside both of the circles.
Notice that “platypodes” is listed in both of the circles. The point of Venn diagrams is that we can show this overlap in set membership by overlapping these circles.
Venn diagrams
Venn diagrams are very useful constructs made of two or more circles that sometimes overlap. Venn diagrams frequently appear in different areas of mathematics but are most common when dealing with sets and probability.
Look at this Venn diagram:
It shows Set A = {1, 5, 6, 7, 8, 9, 10, 12} and Set B = {2, 3, 4, 6, 7, 9, 11, 12, 13}
If we look at the overlapping section of the Venn diagram, this represents A ∩ B = {6, 7, 9, 12} (The intersection of A and B). This contains the numbers that are in both Set A and Set B.
Taking the two circles in their entirety gives us A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} (The union of A and B).
Two sets
The Venn diagram below is the graphical representation of two more sets. The sets represent information about two sisters – Leah (L) and Kelly (K) and their interests.
- We could write the sets as L = {read, play netball, draw} and K = {dance, skate, listen to music}.
- From the diagram, we see that there is no intersection (L ∩ K = {}) meaning that they have no interests in common.
- The union of these two sets would be the set containing the interests of Leah and Kelly:
- L ∪ K = {read, play netball, draw, dance, skate, listen to music}
Try answering the questions below:
Question
List the items in:
- Set A = {12, 14, 15, 16, 17, 18, 20, 22}
- Set B = {16, 17, 20, 21, 22, 23, 24, 25, 28}
Question
List the intersection and union of the following Venn diagram:
Intersection – A ∩ B = {3, 7, 9, 20}
Union – A ∪ B = {3, 7, 9, 10, 14, 15, 19, 20, 23, 24, 25, 26, 30}
What is a Venn Diagram
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A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Often, they serve to graphically organize things, highlighting how the items are similar and different.
Venn diagrams, also called Set diagrams or Logic diagrams, are widely used in mathematics, statistics, logic, teaching, linguistics, computer science and business.
Many people first encounter them in school as they study math or logic, since Venn diagrams became part of “new math” curricula in the 1960s.
These may be simple diagrams involving two or three sets of a few elements, or they may become quite sophisticated, including 3D presentations, as they progress to six or seven sets and beyond.
They are used to think through and depict how items relate to each within a particular “universe” or segment. Venn diagrams allow users to visualize data in clear, powerful ways, and therefore are commonly used in presentations and reports. They are closely related to Euler diagrams, which differ by omitting sets if no items exist in them. Venn diagrams show relationships even if a set is empty.
Venn diagrams are named after British logician John Venn. He wrote about them in an 1880 paper entitled “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings” in the Philosophical Magazine and Journal of Science.
But the roots of this type of diagram go back much further, at least 600 years.
In the 1200s, philosopher and logician Ramon Llull (sometimes spelled Lull) of Majorca used a similar type of diagram, wrote author M.E. Baron in a 1969 article tracing their history.
She also credited German mathematician and philosopher Gottfried Wilhelm von Leibnitz with drawing similar diagrams in the late 1600s.
In the 1700s, Swiss mathematician Leonard Euler (pronounced Oy-ler) invented what came to be known as the Euler Diagram, the most direct forerunner of the Venn Diagram.
In fact, John Venn referred to his own diagrams as Eulerian Circles, not Venn Diagrams. The term Venn Diagrams was first published by American philosopher Clarence Irving (C.I.
) Lewis in his 1918 book, A Survey of Symbolic Logic.
How to Use a Venn Diagram
A Venn diagram is an illustration that uses circles to show the relationships among things or finite groups of things. Circles that overlap have a commonality while circles that do not overlap do not share those traits.
Venn diagrams help to visually represent the similarities and differences between two concepts. They have long been recognized for their usefulness as educational tools. Since the mid-20th century, Venn diagrams have been used as part of the introductory logic curriculum and in elementary-level educational plans around the world.
- A Venn diagram uses circles that overlap or don't overlap to show the commonalities and differences among things or groups of things.
- Things that have commonalities are shown as overlapping circles while things that are distinct stand alone.
- Venn diagrams are now used as illustrations in business and in many academic fields.
The English logician John Venn popularized the diagram in the 1880s. He called them Eulerian circles after the Swiss mathematician Leonard Euler, who created similar diagrams in the 1700s.
The term Venn diagram did not appear until 1918 when Clarence Lewis, an American academic philosopher and the eventual founder of conceptual pragmatism, referred to the circular depiction as the Venn diagram in his book A Survey of Symbolic Logic.
Venn diagrams have been used since the mid-20th century in classrooms from the elementary school level to introductory logic.
Venn studied and taught logic and probability theory at Cambridge University, where he developed his method of using diagrams to illustrate the branch of mathematics known as set theory.
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