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4.3: Venn Diagrams

  • Page ID
    30260
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    Learning Objectives
    • Use a Venn diagram to illustrate the relationship between 2 and 3 sets
    • Create an expression relating 2 and 3 sets from 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 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 \(\PageIndex{1}\)

    Create Venn diagrams to illustrate \(A \cup B, A \cap B,\) and \(A' \cap B\).

    Solution

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

    clipboard_ee03de8fb983d8e26d6f038c378cb00be.png

    \(A \cap B\) contains only those elements in both sets - in the overlap of the circles.

    clipboard_e851dd1ad67dd86f3e5fe2710c0936089.png

    \(A' \cap B\) contains the elements of \(B\) that are not in \(A\).

    4.3Venn.png

    Example \(\PageIndex{2}\)

    Use a Venn diagram to illustrate \((H \cap F)' \cap W\).

    Solution

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

    clipboard_ebe59da114cdbdec91c71a3e8167fd89c.png

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

    clipboard_e5bcfbdfa49842a14d119067dd23ef8cb.png

    Example \(\PageIndex{3}\)

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

    clipboard_e7f0f9dcccbf62d610ee286fa70153e21.png

    Solution

    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 \(\PageIndex{1}\)

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

    clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png

    Answer

    \( \left( A \cup B \right) \cap C'\)


    This page titled 4.3: Venn Diagrams is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.