Probability and Statistics > Probability > How to Draw a Venn Diagram

**Contents:**

Venn diagrams are a way to visualize relationships between groups of objects. They are often confused with Euler diagrams. While both have circles, Venn diagrams show the** whole of a set **while Euler diagrams can show **parts of a set**. Venn diagrams can have unlimited circles, but more than three becomes extremely complicated so you’ll usually see just two or three circles in a Venn diagram drawing.

## Draw a Basic Venn Diagram.

**Sample question**: Draw a Venn diagram to show categories of “indoor” and “outdoor” for the following pets:

Cat, Goat, Rabbit, Tortoise, Hamster, Fish, Horse, Parrot.

Step 1: **Categorize your items** (in this case, pets):

Indoor pets: Hamster, Fish, Parrot.

Outdoor pets: Goat, Tortoise, Horse.

Both categories (indoor *and *outdoor): Cat, Rabbit.

Step 2: **Draw a rectangle and label it**. For this sample question, label the rectangle “Pets.”

Step 3: **Draw two or three circles according to how many categories you have**. There are two categories in the sample question: “indoor pets” and “outdoor pets”, so draw two circles. Make sure the circles overlap.

Step 4: **Write your items in the relevant circle**. If items fit both categories, write those where the circles overlap (the “intersection”).

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**Tip**: If you had a pet that doesn’t fit a category (a virtual pet, for example, might not fit either category), write it within the rectangle but outside the circles.

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## Draw a Venn Diagram of A ∩ B: Overview

An **Intersection (∩)** in Venn diagrams is when items appear in both categories presented (not one or the other). The intersection is written as A ∪ B where A is one category and B is a second category. The ∪ means “intersection,” which in the example above were the pets that were *both indoor and outdoor* : cats and rabbits. It’s written as:

Indoor ∩ Outdoor = {cats,rabbits}.

Which way does the ∪ go? An easy way to remember is that the **union is the ∪** (so by default, the intersection is the other one!).

## Draw a Venn Diagram of A ∩ B: Steps

**Sample question**: Draw a Venn Diagram of A ∩ B (Intersection) of “black things” and “white things”:

Coal, Cats, Tar, Onyx, Snow, milk, shaving cream, hair

Step 1: **Categorize your items** (in this case, “things that are black” and “things that are white”):

Black: Coal, Tar, Onyx

White: Snow, Milk, Shaving Cream

Can be both: Cats, hair

Step 2: **Draw a rectangle and label it**. For this sample question, label the rectangle “Black and/or white things.”

Step 3: **Draw two or three circles according to how many categories you have**. There are two categories in the sample question: “black” and “white”, so draw two circles. Make sure the circles overlap.

Step 4: **Write the items that are in the intersection {cats, hair] **in the space where the circles overlap.

*That’s it!*

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## Draw a Venn Diagram for A ∪ B : Overview

A **Union (∪)** is when items are in **some** or **all** categories. For example, let’s say you wanted to make a Venn diagram for categories of “black things” and “white things”. Your list of items is: Coal, Cats, Tar, Onyx, Snow, milk, shaving cream, and hair.

A union is written (the “mathy way”) as:

Black ∪ White = {Coal, Cats, Tar, Onyx, Snow, milk, shaving cream, hair}. The items that are a union (black, white, or could be both) are all of the items listed: Coal, Cats, Tar, Onyx, Snow, milk, shaving cream, hair.

But how would you put that in a diagram? The answer is to use circles to display each category; the circles overlap where the categories overlap.

**Sample problem**: Draw a Venn Diagram for A ∪ B (Union) for household gadgets that are electronic and/or non-electronic:

Vacuum, Mop, Television, Duster, Toothbrush, Hairdryer, Lawn Mower.

Step 1: **Categorize your items** (in this case, household gadgets):

Electronic: Vacuum, Television, Hairdryer.

Non-Electronic: Mop, Duster.

Both: Toothbrush, Lawn Mower.

Step 2: **Draw a rectangle and label it**. For this sample problem, label it “Household gadgets.”

Step 3: **Draw circles according to how many categories you have**. There are two categories in the sample problem: “electronic” and “non-electronic”, so draw two circles. Make sure the circles overlap.

Step 4: **Write the items in the relevant circle**. If items fit both categories, write those where the circles overlap.

*That’s it!*

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