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Mathematics LibreTexts

2: 1. Introduction to Sets, Venn Diagrams, and Partitions

  • Page ID
    25687
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    Contents:

    1. Sets, elements, set-builder notation
    2. Subsets
    3. Unions, intersections, and the empty set
    4. Complements
    5. Cartesian Products
    6. Introduction to Venn diagrams and shading sets
    7. Determining which set are shaded in a Venn diagram
    8. Definition and examples of partitions
    9. The number of elements in a set: notation, examples, and cartesian products
    10. The number of elements in a set: partitions and an example

     

     

     

     

     

     

     

     

     

     

     

    PREWORK:

    1. Let \(A=\{e,n,o,u,g,h\}\) and \(B=\{s,n,o,w\}\). Determine \(A\cap B\) and \(A\cup B\).
    2. Let \(U\) be the set of all Saint Mary's students. Let \(A\) be the set of Saint Mary's students who live on campus and let \(B\) be the set of Saint Mary's Students who are athletes. Describe the set \(A^c\) in words. Then describe the set \(A\cap B\) in words.
    3. Draw a Venn diagram with three sets \(A\), \(B\), and \(C\) and shade in the area representing \(A\cap B^c\cap C\).
    4. Let \(X=\{1,2,3,4,5,6,7,8,9,10\}\). Let \(A=\{2,3,8\}\), \(B=\{1,3,7,9\}\), and \(C=\{3,4,6,10\}\). Does \(\{A,B,C\}\) form a partition of \(X\)? Explain.

    Solutions:

    1. \(A\cap B=\{n,o\}\) since only n and o appear in both sets.  \(A\cup B=\{e,n,o,u,g,h,s,w\}\) since we take all elements in either set, but get rid of repeats.
    2. \(A^c\) is the set of Saint Mary's students who live off campus, since \(A^c\) contains all elements of \(U\) not in \(A\). \(A\cap B\) is the set of Saint Mary's students who live on campus and are student athletes, since the intersection includes elements that are in both \(A\) and \(B\).
    3. Prework1#3.jpg We need to shade all parts of the Venn diagram that are in \(A\) and not in \(B\) and in \(C\).
    4. No. To form a partition of \(X\) we would need \(A\cup B\cup C=X\) but none of them contain 5. Also we need no element to appear in more than one of \(A\), \(B\), and \(C\), but 3 appears more than once.