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Math Help - Set problem

  1. #1
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    Set problem

    Hello I need help with this problem

    Let
    S be a set with n elements.

    (a)How many ordered pairs are there in
    S S ?
    (b) How many relations are there on
    S (subsets of S S) ?


    Let
    A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,

    |
    A \ C|
    = 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

    \ means intersection and [ means union

    Thank you

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  2. #2
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    Quote Originally Posted by qwerty321 View Post
    Hello I need help with this problem

    Let
    S be a set with n elements.

    (a)How many ordered pairs are there in
    S S ?
    (b) How many relations are there on
    S (subsets of S S) ?


    Let
    A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,

    |
    A \ C|
    = 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

    \ means intersection and [ means union

    Thank you

    (a) If  (x, y) \in S \times S, since there are n elements in S, you have n choices for x and n choices for y. So n*n = ?
    (b) Sorry, can't help you with this one.
    (c) Suggestion: Draw a Venn Diagram.
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  3. #3
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    Quote Originally Posted by qwerty321 View Post
    Let [/FONT]A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,[/LEFT]
    |
    A \ C|
    = 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

    \ means intersection and [ means union

    Thank you

    Use this \left| {A \cup B \cup C} \right| = \left| A \right| + \left| B \right| + \left| C \right| - \left| {A \cap B} \right| - \left| {A \cap C} \right| - \left| {B \cap C} \right| + \left| {A \cap B \cap C} \right|
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  4. #4
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    from where u got the formula?
    thank you
    and could you help me with the question for the relation?
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  5. #5
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    Quote Originally Posted by qwerty321 View Post
    from where u got the formula?
    That is known as inclusion/exclusion. It is one of the most important counting tools.

    Quote Originally Posted by qwerty321 View Post
    (b) How many relations are there on S
    \begin{gathered}<br />
  \left| S \right| = n \hfill \\<br />
  \left| {S \times S} \right| = n^2  \hfill \\<br />
  \left| {\mathbb{P}(S \times S)} \right| = 2^{n^2 }  \hfill \\ <br />
\end{gathered}
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