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7.3: Venn Diagrams

  • Page ID
    58739
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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 illustrations now called Venn Diagrams.

    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 9

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

    A Venn diagram showing two overlapping sets A and B. The region included in either set is highlighted.\(A \cup B\) contains all elements in either set.

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

    A Venn diagram showing two overlapping sets A and B. The overlapping region included in both sets is highlighted.

    A Venn diagram showing two overlapping sets A and B. The region included only in B but not A is highlighted.\(A \cap B\) contains only those elements in both sets - in the overlap of the circles.

    Example 10

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

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

    A Venn diagram of three sets H F and W are shown overlapping.  The region where H and F overlap is highlighted.

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

    A Venn diagram of three sets H F and W are shown overlapping.  The region in set W is highlighted, with the exception of the part that also lies in both F and H.  In other words, all of W, except for the part where all three overlap.

    Example 11

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

    A Venn diagram of three sets H F and W are shown overlapping.  The region where H and F overlap, but W does not, is highlighted.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 \(H \cap F \cap W\)

    Try it Now 3

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

    A Venn diagram of three sets A B and C are shown overlapping.  The region highlighted includes anything in A or anything in B, excluding anything also in C

    Answer

    \(A \cup B \cap C^{c}\)


    This page titled 7.3: Venn Diagrams is shared under a CC BY-SA license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) .

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