venn diagram of thesis statement and topic sentence

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5.2: Identifying Thesis Statements and Topic Sentences

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Being able to identify the purpose and thesis of a text, as you’re reading it, takes practice. This section will offer you that practice.

One fun strategy for developing a deeper understanding the material you’re reading is to make a visual “map” of the ideas. Mind maps, whether hand-drawn or done through computer programs, can be fun to make, and help put all the ideas of an essay you’re reading in one easy-to-read format.

Your understanding of what the “central” element of the mind map is might change as you read and re-read. Developing the central idea of your mind map is a great way to help you determine the reading’s thesis.

Figure 2.5. 1

- Hand-drawn Mind Map

Locating Explicit and Implicit Thesis Statements

In academic writing, the thesis is often explicit : it is included as a sentence as part of the text. It might be near the beginning of the work, but not always–some types of academic writing leave the thesis until the conclusion.

Journalism and reporting also rely on explicit thesis statements that appear very early in the piece–the first paragraph or even the first sentence.

Works of literature, on the other hand, usually do not contain a specific sentence that sums up the core concept of the writing. However, readers should finish the piece with a good understanding of what the work was trying to convey. This is what’s called an implicit thesis statement: the primary point of the reading is conveyed indirectly, in multiple locations throughout the work. (In literature, this is also referred to as the theme of the work.)

Academic writing sometimes relies on implicit thesis statements, as well.

This video offers excellent guidance in identifying the thesis statement of a work, no matter if it’s explicit or implicit.

Topic Sentences

We’ve learned that a thesis statement conveys the primary message of an entire piece of text. Now, let’s look at the next level of important sentences in a piece of text: topic sentences in each paragraph.

A useful metaphor would be to think of the thesis statement of a text as a general: it controls all the major decisions of the writing. There is only one thesis statement in a text. Topic sentences, in this relationship, serve as captains: they organize and sub-divide the overall goals of a writing into individual components. Each paragraph will have a topic sentence.

Figure 2.5. 2

It might be helpful to think of a topic sentence as working in two directions simultaneously. It relates the paragraph to the essay’s thesis, and thereby acts as a signpost for the argument of the paper as a whole, but it also defines the scope of the paragraph itself. For example, consider the following topic sentence:

Many characters in Lorraine Hansberry’s play A Raisin in the Sun have one particular dream in which they are following, though the character Walter pursues his most aggressively.

If this sentence controls the paragraph that follows, then all sentences in the paragraph must relate in some way to Walter and the pursuit of his dream.

Topic sentences often act like tiny thesis statements. Like a thesis statement, a topic sentence makes a claim of some sort. As the thesis statement is the unifying force in the essay, so the topic sentence must be the unifying force in the paragraph. Further, as is the case with the thesis statement, when the topic sentence makes a claim, the paragraph which follows must expand, describe, or prove it in some way. Topic sentences make a point and give reasons or examples to support it.

 The following diagram illustrates how a topic sentence can provide more focus to the general topic at hand.

Placement of Topic Sentences

What if I told you that the topic sentence doesn’t necessarily need to be at the beginning? This might be contrary to what you’ve learned in previous English or writing classes, and that’s okay. Certainly, when authors announce a topic clearly and early on in a paragraph, their readers are likely to grasp their idea and to make the connections that they want them to make.

However, when authors are writing for a more sophisticated academic audience—that is an audience of college-educated readers—they will often use more sophisticated organizational strategies to build and reveal ideas in their writing. One way to think about a topic sentence, is that it presents the broadest view of what authors want their readers to understand. This is to say that they’re providing a broad statement that either announces or brings into focus the purpose or the meaning for the details of the paragraph. If the topic sentence is seen as the broadest view, then every supporting detail will bring a narrower—or more specific—view of the same topic.

With this in mind, take some time to contemplate the diagrams in the figure below. The widest point of each diagram (the bases of the triangles) represents the topic sentence of the paragraph. As details are presented, the topic becomes narrower and more focused. The topic can precede the details, it can follow them, it can both precede and follow them, or the details can surround the topic. There are surely more alternatives than those that are presented here, but this gives you an idea of some of the possible paragraph structures and possible placements for the topic sentence of a paragraph.

Consider some of the following examples of different topic sentence placements in a paragraph from a review essay of the beloved children’s book, The Cat in the Hat , by Dr. Seuss. Paragraph structures are labeled according to the diagrams presented above, and topic sentences are identified by red text.

Topic Sentence-Details-Topic Sentence

A good children’s book requires an exciting plot and a problem with which children can sympathize. In The Cat in the Hat there is plenty of action, depicted in the wild antics of the cat, and later in the amazing but dangerous and messy tricks of Thing 1 and Thing 2. All this excitement and action naturally draws children into the story and keeps the plot moving forward at a pace that maintains their interest. There is also tension to be resolved. The fish senses danger and constantly warns the children not to participate in the cat’s perilous stunts. And later, as the mother’s return becomes more imminent, the children begin to heed the fish’s warning and finally wish to contain the chaos and clean up the mess, but how? While this plot is fantastic enough to fuel any child’s imagination, it also contains a problem with which any child can relate: a mess and the threat of a parent’s disapproval. The careful balance of action, tension, and relatability is what makes this book an enduring childhood favorite.

Topic Sentence-Details

The careful balance of action, tension, and relatability is what makes Dr. Seuss’s The Cat in the Hat an enduring childhood favorite. In The Cat in the Hat there is plenty of action, depicted in the wild antics of the cat, and later in the amazing but dangerous and messy tricks of Thing 1 and Thing 2. All this excitement and action naturally draws children into the story and keeps the plot moving forward at a pace that maintains their interest. There is also tension to be resolved. The fish senses danger and constantly warns the children not to participate in the cat’s perilous stunts. And later, as the mother’s return becomes more imminent, the children begin to heed the fish’s warning and finally wish to contain the chaos and clean up the mess, but how? While this plot is fantastic enough to fuel any child’s imagination, it also contains a problem with which any child can relate: a mess and the threat of a parent’s disapproval.

You can relocate the topic sentence to the end here, and you’ll have an example of the Details-Topic Sentence method of organizing the paragraph.

Details-Topic Sentence-Details

In The Cat in the Hat there is plenty of action, depicted in the wild antics of the cat, and later in the amazing but dangerous and messy tricks of Thing 1 and Thing 2. All this excitement and action naturally draws children into the story and keeps the plot moving forward at a pace that maintains their interest. The careful balance of action, tension, and relatability is what makes Dr. Seuss’s The Cat in the Hat an enduring childhood favorite. There is definitely tension to be resolved here. The fish senses danger and constantly warns the children not to participate in the cat’s perilous stunts. And later, as the mother’s return becomes more imminent, the children begin to heed the fish’s warning and finally wish to contain the chaos and clean up the mess, but how? While this plot is fantastic enough to fuel any child’s imagination, it also contains a problem with which any child can relate: a mess and the threat of a parent’s disapproval.

Explicit and Implicit Topic Sentences

Similar to thesis statements, topic sentences may be explicit or implicit.

Consider the following paragraph from an essay titled “The Bothersome Beauty of Pigeons,” by author and Boise State writing professor, Bruce Ballenger. It’s important to note that this is a personal narrative essay rather than a more traditional academic essay, but the paragraph provides a good example of an implied topic. In this essay Ballenger takes the time to consider the beauty of pigeons, a bird that’s usually thought of as nothing more than a nuisance. Just prior to this paragraph, Ballenger talks about how he used a fake owl to scare away pigeons on his property. He goes on to explain,

My pigeons moved next door where an elderly couple feed them bird seed and have the time and willingness to clean up after their new charges; so it seems, in this case, things have worked out for everyone. But the large flocks still haunt the piazzas in Florence and Venice, the squares in London, and similar places in nearly every city across the globe. Despite their ability to distinguish between a Van Gogh and a Chagall, pigeons still deposit droppings that deface the great marble statues and facades–the works of art and architecture that are part of our human heritage–and yet people still buy bags of seed for about a dollar and pose for photographs, drenched in doves. Meanwhile, officials in these cities continue, sometimes quietly, to wage war against the birds (“Introduction”).

Here, Ballenger seems to be saying that in spite of the attempts of so many to rid themselves of the pigeons, others are still drawn to them and will feed them and encourage them to come back. His main idea seems to be that the battle against pigeons is a losing proposition, but he doesn’t come out and say so. His message in this paragraph is implied. Do you think this paragraph would be improved with an explicit topic sentence?

EXERCISE 1: Identify the Topic and Focus

Choose a piece of writing, perhaps an essay or some news articles found online, and for each paragraph identify (1) the topic and (2) the more focused idea. Remember, the topic sentence applies more focus to the broader topic to help narrow the scope of the paragraph. For example, the topic of a paragraph might be school lunches. The more focused idea of that same paragraph might be the idea of having students plant school gardens as a way to help incorporate more fresh produce in the menu.

License and Attributions:

CC licensed content, Previously shared:

Basic Reading and Writing. Authored by: Lumen. Located at:  https://human.libretexts.org/Bookshelves/Composition/Book:_Basic_Reading_and_Writing_(Lumen)/ Module_2:_Critical_Reading/2.05:_Identifying_Thesis_Statements   License: CC BY: Attribution.

Adaptions: Reformatted, some content removed to fit a broader audience.

Difference Between Topic Sentence Vs. Thesis Statement

After being introduced during the K-12 years of a student's education, academic writing is a mainstay in college and higher education. With proper format and technique being a major aspect of academic writing, it is essential for students to understand the differences between a topic sentence and a thesis statement.

Explore this article

• Essay Format
• Topic Sentences
• Thesis Statement
• Differences

1 Essay Format

Before delving into the differences between topic sentences and thesis statements, it is important to first establish the general format for a written essay. For K-12 students, written essays can commonly require the five-paragraph format, which consist of an introduction, three paragraphs that make up the body of the paper and the final paragraph as the conclusion. By understanding this general format of academic writing, it is easier to understand where topic sentences and thesis statements fit into place.

2 Topic Sentences

Topic sentences are used in each major paragraph of an essay paper. The purposes of these sentences are to develop and state the key idea of each paragraph in an effort to define the point of the paragraph. For example, when writing an essay paper on the effects of the Gettysburg Address, one of the paragraphs in the body of the paper could discuss how the audience received President Lincoln's famous speech. The topic sentence for this paragraph could be, "With the brief yet poignant nature of the Gettysburg Address, President Lincoln reminded his audience why the country was fighting so desperately to conserve the Union."

3 Thesis Statement

A thesis statement is a single sentence that is used to define the purpose of your paper. Thesis statements inform readers as to the purpose of your paper. After reading your thesis statement, anyone reviewing your paper should be able to determine what your paper is going to prove or argue. For example, a thesis statement for a paper about William Shakespeare's famous play "Romeo and Juliet" could be as follows: "The play 'Romeo and Juliet' demonstrates the incredible power and impatient nature of young love, as the star-crossed lovers' inability to exercise patience or control their passion for one another results in their deaths."

4 Differences

With the purposes of both topic sentences and thesis statements established, the differences between the two are clear. Thesis statements need to be more broad and leave room to expand on further points to support the paper's argument. With the thesis statement in the previous section, a writer could continue explaining his argument about Romeo and Juliet by providing sub-points, demonstrating instances where the pair made hasty choices that had fatal consequences. Topic sentences are more specific in their purpose. For example, with the topic sentence in section two, it is clear that the next paragraph will discuss how President Lincoln used the Gettysburg Address to remind his countrymen about the purpose of the Civil War.

• 1 Indiana University: Paragraphs and Topic Sentences

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Comparing and Contrasting

What this handout is about.

This handout will help you first to determine whether a particular assignment is asking for comparison/contrast and then to generate a list of similarities and differences, decide which similarities and differences to focus on, and organize your paper so that it will be clear and effective. It will also explain how you can (and why you should) develop a thesis that goes beyond “Thing A and Thing B are similar in many ways but different in others.”

Introduction

In your career as a student, you’ll encounter many different kinds of writing assignments, each with its own requirements. One of the most common is the comparison/contrast essay, in which you focus on the ways in which certain things or ideas—usually two of them—are similar to (this is the comparison) and/or different from (this is the contrast) one another. By assigning such essays, your instructors are encouraging you to make connections between texts or ideas, engage in critical thinking, and go beyond mere description or summary to generate interesting analysis: when you reflect on similarities and differences, you gain a deeper understanding of the items you are comparing, their relationship to each other, and what is most important about them.

Recognizing comparison/contrast in assignments

Some assignments use words—like compare, contrast, similarities, and differences—that make it easy for you to see that they are asking you to compare and/or contrast. Here are a few hypothetical examples:

• Compare and contrast Frye’s and Bartky’s accounts of oppression.
• Compare WWI to WWII, identifying similarities in the causes, development, and outcomes of the wars.
• Contrast Wordsworth and Coleridge; what are the major differences in their poetry?

Notice that some topics ask only for comparison, others only for contrast, and others for both.

But it’s not always so easy to tell whether an assignment is asking you to include comparison/contrast. And in some cases, comparison/contrast is only part of the essay—you begin by comparing and/or contrasting two or more things and then use what you’ve learned to construct an argument or evaluation. Consider these examples, noticing the language that is used to ask for the comparison/contrast and whether the comparison/contrast is only one part of a larger assignment:

• Choose a particular idea or theme, such as romantic love, death, or nature, and consider how it is treated in two Romantic poems.
• How do the different authors we have studied so far define and describe oppression?
• Compare Frye’s and Bartky’s accounts of oppression. What does each imply about women’s collusion in their own oppression? Which is more accurate?
• In the texts we’ve studied, soldiers who served in different wars offer differing accounts of their experiences and feelings both during and after the fighting. What commonalities are there in these accounts? What factors do you think are responsible for their differences?

You may want to check out our handout on understanding assignments for additional tips.

Using comparison/contrast for all kinds of writing projects

Sometimes you may want to use comparison/contrast techniques in your own pre-writing work to get ideas that you can later use for an argument, even if comparison/contrast isn’t an official requirement for the paper you’re writing. For example, if you wanted to argue that Frye’s account of oppression is better than both de Beauvoir’s and Bartky’s, comparing and contrasting the main arguments of those three authors might help you construct your evaluation—even though the topic may not have asked for comparison/contrast and the lists of similarities and differences you generate may not appear anywhere in the final draft of your paper.

Discovering similarities and differences

Making a Venn diagram or a chart can help you quickly and efficiently compare and contrast two or more things or ideas. To make a Venn diagram, simply draw some overlapping circles, one circle for each item you’re considering. In the central area where they overlap, list the traits the two items have in common. Assign each one of the areas that doesn’t overlap; in those areas, you can list the traits that make the things different. Here’s a very simple example, using two pizza places:

To make a chart, figure out what criteria you want to focus on in comparing the items. Along the left side of the page, list each of the criteria. Across the top, list the names of the items. You should then have a box per item for each criterion; you can fill the boxes in and then survey what you’ve discovered.

Here’s an example, this time using three pizza places:

As you generate points of comparison, consider the purpose and content of the assignment and the focus of the class. What do you think the professor wants you to learn by doing this comparison/contrast? How does it fit with what you have been studying so far and with the other assignments in the course? Are there any clues about what to focus on in the assignment itself?

Here are some general questions about different types of things you might have to compare. These are by no means complete or definitive lists; they’re just here to give you some ideas—you can generate your own questions for these and other types of comparison. You may want to begin by using the questions reporters traditionally ask: Who? What? Where? When? Why? How? If you’re talking about objects, you might also consider general properties like size, shape, color, sound, weight, taste, texture, smell, number, duration, and location.

Two historical periods or events

• When did they occur—do you know the date(s) and duration? What happened or changed during each? Why are they significant?
• What kinds of work did people do? What kinds of relationships did they have? What did they value?
• What kinds of governments were there? Who were important people involved?
• What caused events in these periods, and what consequences did they have later on?

Two ideas or theories

• What are they about?
• Did they originate at some particular time?
• Who created them? Who uses or defends them?
• What is the central focus, claim, or goal of each? What conclusions do they offer?
• How are they applied to situations/people/things/etc.?
• Which seems more plausible to you, and why? How broad is their scope?
• What kind of evidence is usually offered for them?

Two pieces of writing or art

• What are their titles? What do they describe or depict?
• What is their tone or mood? What is their form?
• Who created them? When were they created? Why do you think they were created as they were? What themes do they address?
• Do you think one is of higher quality or greater merit than the other(s)—and if so, why?
• For writing: what plot, characterization, setting, theme, tone, and type of narration are used?
• Where are they from? How old are they? What is the gender, race, class, etc. of each?
• What, if anything, are they known for? Do they have any relationship to each other?
• What are they like? What did/do they do? What do they believe? Why are they interesting?
• What stands out most about each of them?

Deciding what to focus on

By now you have probably generated a huge list of similarities and differences—congratulations! Next you must decide which of them are interesting, important, and relevant enough to be included in your paper. Ask yourself these questions:

• What’s relevant to the assignment?
• What’s relevant to the course?
• What’s interesting and informative?
• What matters to the argument you are going to make?
• What’s basic or central (and needs to be mentioned even if obvious)?
• Overall, what’s more important—the similarities or the differences?

Suppose that you are writing a paper comparing two novels. For most literature classes, the fact that they both use Caslon type (a kind of typeface, like the fonts you may use in your writing) is not going to be relevant, nor is the fact that one of them has a few illustrations and the other has none; literature classes are more likely to focus on subjects like characterization, plot, setting, the writer’s style and intentions, language, central themes, and so forth. However, if you were writing a paper for a class on typesetting or on how illustrations are used to enhance novels, the typeface and presence or absence of illustrations might be absolutely critical to include in your final paper.

Sometimes a particular point of comparison or contrast might be relevant but not terribly revealing or interesting. For example, if you are writing a paper about Wordsworth’s “Tintern Abbey” and Coleridge’s “Frost at Midnight,” pointing out that they both have nature as a central theme is relevant (comparisons of poetry often talk about themes) but not terribly interesting; your class has probably already had many discussions about the Romantic poets’ fondness for nature. Talking about the different ways nature is depicted or the different aspects of nature that are emphasized might be more interesting and show a more sophisticated understanding of the poems.

Your thesis

The thesis of your comparison/contrast paper is very important: it can help you create a focused argument and give your reader a road map so they don’t get lost in the sea of points you are about to make. As in any paper, you will want to replace vague reports of your general topic (for example, “This paper will compare and contrast two pizza places,” or “Pepper’s and Amante are similar in some ways and different in others,” or “Pepper’s and Amante are similar in many ways, but they have one major difference”) with something more detailed and specific. For example, you might say, “Pepper’s and Amante have similar prices and ingredients, but their atmospheres and willingness to deliver set them apart.”

Be careful, though—although this thesis is fairly specific and does propose a simple argument (that atmosphere and delivery make the two pizza places different), your instructor will often be looking for a bit more analysis. In this case, the obvious question is “So what? Why should anyone care that Pepper’s and Amante are different in this way?” One might also wonder why the writer chose those two particular pizza places to compare—why not Papa John’s, Dominos, or Pizza Hut? Again, thinking about the context the class provides may help you answer such questions and make a stronger argument. Here’s a revision of the thesis mentioned earlier:

Pepper’s and Amante both offer a greater variety of ingredients than other Chapel Hill/Carrboro pizza places (and than any of the national chains), but the funky, lively atmosphere at Pepper’s makes it a better place to give visiting friends and family a taste of local culture.

You may find our handout on constructing thesis statements useful at this stage.

Organizing your paper

There are many different ways to organize a comparison/contrast essay. Here are two:

Subject-by-subject

Begin by saying everything you have to say about the first subject you are discussing, then move on and make all the points you want to make about the second subject (and after that, the third, and so on, if you’re comparing/contrasting more than two things). If the paper is short, you might be able to fit all of your points about each item into a single paragraph, but it’s more likely that you’d have several paragraphs per item. Using our pizza place comparison/contrast as an example, after the introduction, you might have a paragraph about the ingredients available at Pepper’s, a paragraph about its location, and a paragraph about its ambience. Then you’d have three similar paragraphs about Amante, followed by your conclusion.

The danger of this subject-by-subject organization is that your paper will simply be a list of points: a certain number of points (in my example, three) about one subject, then a certain number of points about another. This is usually not what college instructors are looking for in a paper—generally they want you to compare or contrast two or more things very directly, rather than just listing the traits the things have and leaving it up to the reader to reflect on how those traits are similar or different and why those similarities or differences matter. Thus, if you use the subject-by-subject form, you will probably want to have a very strong, analytical thesis and at least one body paragraph that ties all of your different points together.

A subject-by-subject structure can be a logical choice if you are writing what is sometimes called a “lens” comparison, in which you use one subject or item (which isn’t really your main topic) to better understand another item (which is). For example, you might be asked to compare a poem you’ve already covered thoroughly in class with one you are reading on your own. It might make sense to give a brief summary of your main ideas about the first poem (this would be your first subject, the “lens”), and then spend most of your paper discussing how those points are similar to or different from your ideas about the second.

Point-by-point

Rather than addressing things one subject at a time, you may wish to talk about one point of comparison at a time. There are two main ways this might play out, depending on how much you have to say about each of the things you are comparing. If you have just a little, you might, in a single paragraph, discuss how a certain point of comparison/contrast relates to all the items you are discussing. For example, I might describe, in one paragraph, what the prices are like at both Pepper’s and Amante; in the next paragraph, I might compare the ingredients available; in a third, I might contrast the atmospheres of the two restaurants.

If I had a bit more to say about the items I was comparing/contrasting, I might devote a whole paragraph to how each point relates to each item. For example, I might have a whole paragraph about the clientele at Pepper’s, followed by a whole paragraph about the clientele at Amante; then I would move on and do two more paragraphs discussing my next point of comparison/contrast—like the ingredients available at each restaurant.

There are no hard and fast rules about organizing a comparison/contrast paper, of course. Just be sure that your reader can easily tell what’s going on! Be aware, too, of the placement of your different points. If you are writing a comparison/contrast in service of an argument, keep in mind that the last point you make is the one you are leaving your reader with. For example, if I am trying to argue that Amante is better than Pepper’s, I should end with a contrast that leaves Amante sounding good, rather than with a point of comparison that I have to admit makes Pepper’s look better. If you’ve decided that the differences between the items you’re comparing/contrasting are most important, you’ll want to end with the differences—and vice versa, if the similarities seem most important to you.

Our handout on organization can help you write good topic sentences and transitions and make sure that you have a good overall structure in place for your paper.

Cue words and other tips

To help your reader keep track of where you are in the comparison/contrast, you’ll want to be sure that your transitions and topic sentences are especially strong. Your thesis should already have given the reader an idea of the points you’ll be making and the organization you’ll be using, but you can help them out with some extra cues. The following words may be helpful to you in signaling your intentions:

• like, similar to, also, unlike, similarly, in the same way, likewise, again, compared to, in contrast, in like manner, contrasted with, on the contrary, however, although, yet, even though, still, but, nevertheless, conversely, at the same time, regardless, despite, while, on the one hand … on the other hand.

For example, you might have a topic sentence like one of these:

• Compared to Pepper’s, Amante is quiet.
• Like Amante, Pepper’s offers fresh garlic as a topping.
• Despite their different locations (downtown Chapel Hill and downtown Carrboro), Pepper’s and Amante are both fairly easy to get to.

You may reproduce it for non-commercial use if you use the entire handout and attribute the source: The Writing Center, University of North Carolina at Chapel Hill

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Venn Diagrams to Plan Essays and More

• M.Ed., Education Administration, University of Georgia
• B.A., History, Armstrong State University

A Venn diagram is a great tool for brainstorming and creating a comparison between two or more objects, events, or people. You can use this as a first step to creating an outline for a  compare and contrast essay .

Simply draw two (or three) large circles and give each circle a title, reflecting each object, trait, or person you are comparing.

Inside the intersection of the two circles (overlapping area), write all the traits that the objects have in common. You will refer to these traits when you  compare  similar characteristics.

In the areas outside the overlapping section, you will write all of the traits that are specific to that particular object or person.

Creating an Outline for Your Essay Using a Venn Diagram

From the Venn diagram above, you can create an easy outline for your paper. Here is the beginning of an essay outline:

1. Both dogs and cats make great pets.

• Both animals can be very entertaining
• Each is loving in its own way
• Each can live inside or outside the house

2. Both have drawbacks, as well.

• They can damage property
• Both can be costly
• Both require time and attention

3. Cats can be easier to care for.

• Leaving for a day

4. Dogs can be better companions.

• Going to the park
• Going for walks
• Will enjoy my company

As you can see, outlining is much easier when you have a visual aid to help you with the brainstorming process.

More Uses for Venn Diagrams

Besides its usefulness for planning essays, Venn Diagrams can be used for thinking through many other problems both at school and at home. For example:

• Planning a Budget: Create three circles for What I Want, What I Need, and What I Can Afford.
• Setting Priorities: Create circles for different types of priorities: School, Chores, Friends, TV, along with a circle for What I Have Time for This Week.
• Choosing Activities: Create circles for different types of activities: What I'm Committed to, What I'd Like to Try, and What I Have Time for Each Week.
• Comparing People's Qualities: Create circles for the different qualities you're comparing (ethical, friendly, good looking, wealthy, etc.), and then add names to each circle. Which overlap?
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• How to Compare Two Novels in Comparative Essay
• Beef Up Critical Thinking and Writing Skills: Comparison Essays
• 101 Compare and Contrast Essay Topics
• 7 Active Reading Strategies for Students
• Using Graphic Organizers for Special Education
• Writing Cause and Effect Essays for English Learners
• Vocabulary Chart ESL Lesson Plan
• Discover Ideas Through Brainstorming
• Graphic Organizers
• The Prewriting Stage of the Writing Process
• What Is a Cladogram? Definition and Examples
• Second Grade Map Project Ideas
• Miss Nelson Is Missing Lesson Plan

1.3 Understanding Venn Diagrams

Learning objectives.

After completing this section, you should be able to:

• Utilize a universal set with two sets to interpret a Venn diagram.
• Utilize a universal set with two sets to create a Venn diagram.
• Determine the complement of a set.

Have you ever ordered a new dresser or bookcase that required assembly? When your package arrives you excitedly open it and spread out the pieces. Then you check the assembly guide and verify that you have all the parts required to assemble your new dresser. Now, the work begins. Luckily for you, the assembly guide includes step-by-step instructions with images that show you how to put together your product. If you are really lucky, the manufacturer may even provide a URL or QR code connecting you to an online video that demonstrates the complete assembly process. We can likely all agree that assembly instructions are much easier to follow when they include images or videos, rather than just written directions. The same goes for the relationships between sets.

Interpreting Venn Diagrams

Venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. Venn diagrams are named after the mathematician John Venn, who first popularized their use in the 1880s. When we use a Venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. The entire set of data under consideration is known as the universal set .

Consider the statement: All trees are plants. This statement expresses the relationship between the set of all plants and the set of all trees. Because every tree is a plant, the set of trees is a subset of the set of plants. 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 . ” Then we draw a circle within the universal set and label it with the word “Trees.”

This section will introduce how to interpret and construct Venn diagrams. In future sections, as we expand our knowledge of relationships between sets, we will also develop our knowledge and use of Venn diagrams to explore how multiple sets can be combined to form new sets.

Example 1.18

Interpreting the relationship between sets in a venn diagram.

Write the relationship between the sets in the following Venn diagram, in words and symbolically.

The set of terriers is a subset of the universal set of dogs. In other words, the Venn diagram depicts the relationship that all terriers are dogs. This is expressed symbolically as T ⊂ U . T ⊂ U .

Your Turn 1.18

So far, the only relationship we have been considering between two sets is the subset relationship, but sets can be related in other ways. Lions and tigers are both different types of cats, but no lions are tigers, and no tigers are lions. Because the set of all lions and the set of all tigers do not have any members in common, we call these two sets disjoint sets , or non-overlapping sets.

Two sets A A and B B are disjoint sets if they do not share any elements in common. That is, if a a is a member of set A A , then a a is not a member of set B B . If b b is a member of set B B , then b b is not a member of set A A . To represent the relationship between the set of all cats and the sets of lions and tigers using a Venn diagram, we draw the universal set of cats as a rectangle and then draw a circle for the set of lions and a separate circle for the set of tigers within the rectangle, ensuring that the two circles representing the set of lions and the set of tigers do not touch or overlap in any way.

Example 1.19

Describing the relationship between sets.

Describe the relationship between the sets in the following Venn diagram.

The set of triangles and the set of squares are two disjoint subsets of the universal set of two-dimensional figures. The set of triangles does not share any elements in common with the set of squares. No triangles are squares and no squares are triangles, but both squares and triangles are 2D figures.

Your Turn 1.19

Creating venn diagrams.

The main purpose of a Venn diagram is to help you visualize the relationship between sets. As such, it is necessary to be able to draw Venn diagrams from a written or symbolic description of the relationship between sets.

To create a Venn diagram:

• Draw a rectangle to represent the universal set, and label it U = set name U = set name .
• Draw a circle within the rectangle to represent a subset of the universal set and label it with the set name.

If there are multiple disjoint subsets of the universal set, their separate circles should not touch or overlap.

Example 1.20

Drawing a venn diagram to represent the relationship between two sets.

Draw a Venn diagram to represent the relationship between each of the sets.

• All rectangles are parallelograms.
• All women are people.
• The set of rectangles is a subset of the set of parallelograms. First, draw a rectangle to represent the universal set and label it with U = Parallelograms U = Parallelograms , then draw a circle completely within the rectangle, and label it with the name of the set it represents, R = Rectangles R = Rectangles .

In this example, both letters and names are used to represent the sets involved, but this is not necessary. You may use either letters or names alone, as long as the relationship is clearly depicted in the diagram, as shown below.

• The universal set is the set of people, and the set of all women is a subset of the set of people.

Your Turn 1.20

Example 1.21, drawing a venn diagram to represent the relationship between three sets.

All bicycles and all cars have wheels, but no bicycle is a car. Draw a Venn diagram to represent this relationship.

Step 1: The set of bicycles and the set of cars are both subsets of the set of things with wheels. The universal set is the set of things with wheels, so we first draw a rectangle and label it with U = Things with Wheels U = Things with Wheels .

Step 2: Because the set of bicycles and the set of cars do not share any elements in common, these two sets are disjoint and must be drawn as two circles that do not touch or overlap with the universal set.

Your Turn 1.21

The complement of a set.

Recall that if set A A is a proper subset of set U U , the universal set (written symbolically as A ⊂ U A ⊂ U ), then there is at least one element in set U U that is not in set A A . The set of all the elements in the universal set U U that are not in the subset A A is called the complement of set A A , A ' A ' . In set builder notation this is written symbolically as: A ' = { x ∈ U | x ∉ A } . A ' = { x ∈ U | x ∉ A } . The symbol ∈ ∈ is used to represent the phrase, “is a member of,” and the symbol ∉ ∉ is used to represent the phrase, “is not a member of.” In the Venn diagram below, the complement of set A A is the region that lies outside the circle and inside the rectangle. The universal set U U includes all of the elements in set A A and all of the elements in the complement of set A A , and nothing else.

Consider the set of digit numbers. Let this be our universal set, U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Now, let set A A be the subset of U U consisting of all the prime numbers in set U U , A = { 2 , 3 , 5 , 7 } . A = { 2 , 3 , 5 , 7 } . The complement of set A A is A ' = { 0 , 1 , 4 , 6 , 8 , 9 } . A ' = { 0 , 1 , 4 , 6 , 8 , 9 } . The following Venn diagram represents this relationship graphically.

Example 1.22

Finding the complement of a set.

For both of the questions below, A A is a proper subset of U U .

• Given the universal set U = { Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran} U = { Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran} and set A = { Donald Glover, Bruno Mars, Ed Sheeran} A = { Donald Glover, Bruno Mars, Ed Sheeran} , find A ' . A ' .
• Given the universal set U = { d|d is a dog } U = { d|d is a dog } and B = { b ∈ U|b is a beagle } B = { b ∈ U|b is a beagle } , find B ' . B ' .
• The complement of set A A is the set of all elements in the universal set U U that are not in set A . A . A ' = { Billie Eilish, Adele } A ' = { Billie Eilish, Adele } .
• The complement of set B B is the set of all dogs that are not beagles. All members of set B ′ B ′ are in the universal set because they are dogs, but they are not in set B , B , because they are not beagles. This relationship can be expressed in set build notation as follows: B ′ = { All dogs that are not beagles .} B ′ = { All dogs that are not beagles .} , B ′ = { d ∈ U | d is not a beagle .} B ′ = { d ∈ U | d is not a beagle .} , or B ′ = { d ∈ U | d ∉ B } . B ′ = { d ∈ U | d ∉ B } .

Your Turn 1.22

Check your understanding, section 1.3 exercises.

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• Authors: Donna Kirk
• Publisher/website: OpenStax
• Book title: Contemporary Mathematics
• Publication date: Mar 22, 2023
• Location: Houston, Texas
• Book URL: https://openstax.org/books/contemporary-mathematics/pages/1-introduction
• Section URL: https://openstax.org/books/contemporary-mathematics/pages/1-3-understanding-venn-diagrams

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