Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

13.3: Venn Diagrams

  • Page ID
    34257
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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_e851dd1ad67dd86f3e5fe2710c0936089.png

    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 \(\mathrm{H} \cap P\)

    clipboard_ebe59da114cdbdec91c71a3e8167fd89c.png

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

    clipboard_e5bcfbdfa49842a14d119067dd23ef8cb.png

    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

    clipboard_ee6f4b6ece2cde3e7320d09311c24e49b.png

    Answer

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


    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.