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8.4: Venn Diagrams

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    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 illustration show called Venn Diagrams.

    Definition: Venn Diagrams

    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 B, A B, and Ac B A B contains all elements in either set.

    Figure 10. Circles A & B

    A B contains only those elements in both sets – in the overlap of the circles.

    Figure 11. Circles A & B

    Ac will contain all elements not in the set A. Ac B will contain the elements in set B that are not in set A.

    Figure 12. Circles A & B

    Example 2

    Use a Venn diagram to illustrate (H F)c W

    We’ll start by identifying everything in the set H F

    Figure 13. Circles H, F, & W

    Now, (H F)c W will contain everything not in the set identified above that is also in set W.

    Figure 14. Circles H, F, & W

    Example 3

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

    The elements in the outlined set are in sets H and F, but are not in set W. So we could represent this set as H F Wc

    Figure 15. Circles H, F, & W

    Try it Now

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

    Figure 16. Circles A, B, & C

    This page titled 8.4: Venn Diagrams is shared under a CC BY license and was authored, remixed, and/or curated by Mehgan Andrade and Neil Walker.

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