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3.1.2: Venn Diagrams

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    74303

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    3.1.2 Learning Objectives

    • Use Venn diagrams to depict unions, intersections, and complements of sets

    • Create an expression to describe a section of a Venn diagram

    To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations are now called Venn Diagrams.

    Definition: Venn Diagram

    A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.

    Basic Venn diagrams can illustrate the interaction of two or three sets.

    Example 1

    Create Venn diagrams to illustrate \(A \cup B, A \cap B,\) and \(A^{c} \cap B\)

    \(A \cap B\) contains all elements in both sets - elements in A and B.

    \(A \cup B\) contains all elements in either set - elements in A or B (or both).

    \(A^{c} \cap B\) contains elements outside of A and also in B.

    clipboard_e851dd1ad67dd86f3e5fe2710c0936089.png          clipboard_ee03de8fb983d8e26d6f038c378cb00be.png           clipboard_e17d1da8e9a0f9f6bec3f62801b88cc45.png

     

    Example 2

    Use a Venn diagram to illustrate \((H \cap F)^{c} \cap W\)

    We'll start by identifying everything in the set \(\mathrm{H} \cap F\)

    clipboard_ebe59da114cdbdec91c71a3e8167fd89c.png

    Now, \((H \cap F)^{c} \cap W\) will contain everything not in the set identified above that is also in set \(W\)

    clipboard_e5bcfbdfa49842a14d119067dd23ef8cb.png

    Example 3

    Create an expression to represent the outlined part of the Venn diagram shown.

    clipboard_e7f0f9dcccbf62d610ee286fa70153e21.png

    The elements in the outlined set are in sets \(\mathrm{H}\) and \(F\), but are not in set \(W\). So we could represent this set as \(H \cap F \cap W^{c}\)

    Sometimes Venn diagrams include numbers to show the cardinality of each region in the diagram.

    Example 4

    Use the Venn Diagram representing the number of people who ordered the given ice cream flavors to answer the following questions.

    triple venn.PNG

    1. How many people ordered chocolate and strawberry?
    2. How many people ordered chocolate or strawberry?
    3. How many people ordered chocolate and strawberry but not vanilla?
    4. How many people ordered only vanilla?
    5. How many people ordered vanilla?
    6. How many people did not order chocolate, vanilla, or strawberry?
    7. How many people ordered all three flavors?
    8. How many total orders were there?

     

     

     

    Solution

    1. Look at the intersection of chocolate and strawberry to see that 13 people ordered that combination.
    2. Look at the union of the chocolate circle with the strawberry circle to see that 79 people ordered chocolate or strawberry.
    3. Look at the intersection of chocolate and strawberry and exclude those who ordered vanilla.
    4. Look at the section of the vanilla circle that does not intersect with any other flavor to see that 6 people ordered only vanilla.
    5. Look at all of the regions inside the vanilla circle to see that 48 people ordered vanilla.
    6. Look outside the Venn diagram to see that 15 orders did not contain any of those flavors.
    7. Look in the center where all three circles overlap to see that 9 people ordered all three flavors.
    8. Add the numbers in each section of the Venn and the number outside the Venn to find that there were a total of 100 orders.

    Try it Now 1

    Create an expression to represent the outlined portion of the Venn diagram shown

    clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png

    Answer

    \(A \cup B \cap C^{c}\)    

    This page titled 3.1.2: Venn Diagrams is shared under a CC BY-SA license and was authored, remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.