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Mathematics LibreTexts

13.3: 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 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 \(A^{c} \cap B\)

    clipboard_ee03de8fb983d8e26d6f038c378cb00be.png\(A \cup B\) contains all elements in either set.

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


    clipboard_e17d1da8e9a0f9f6bec3f62801b88cc45.png\(A \cap B\) 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 \(\mathrm{H} \cap P\)


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


    Example 11

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

    clipboard_e7f0f9dcccbf62d610ee286fa70153e21.pngThe 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



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

    13.3: Venn Diagrams is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.