Chapter 5 Sets
5.3 Understanding Venn Diagrams
Learning Objectives
By the end of this section, you will 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 [latex]\text{Trees} \subset \text{Plants}[/latex].
To create a Venn diagram, first we draw a rectangle and label the universal set [latex]\text{“U = Plants”}[/latex]. Then we draw a circle within the universal set and label it with the word [latex]\text{“Trees.”}[/latex]
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
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 [latex]T \subset U[/latex].
Exercise 1
Write the relationship between the sets in the following Venn diagram, in words and symbolical