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Thread: Set Cardinalities

  1. #1
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    Set Cardinalities

    Hey guys, I need help with this problem.

    In a regular deck of playing cards, there are 26 red cards and 12 face cards. Explain, using sets and cardinalities, why there are only 32 cards which are either red or a face card.
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  2. #2
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    Re: Set Cardinalities

    let

    $R$ be the set of red cards

    $F$ be the set of face cards

    first off we know that

    $|R|=26$

    $|F|=12$

    $|R \cap F| = \text{# of red face cards} = 6$

    then via set theory we know for any two sets $A$ and $B$

    $|A \cup B| = |A| + |B| - |A \cap B|$

    and thus

    $\text{size of the set of cards that are either red or a face card} =|R \cup F| = |R|+|F|-|R \cap F| = 26 + 12 - 6 = 32$
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  3. #3
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    Re: Set Cardinalities

    Or: every suit contains 13 cards, 3 of them face cards, 10 not face cards. So "hearts" and "diamonds" contain 2*10= 20 red cards that are not face cards. There are 4*3= 12 face cards so there are a total of 20+ 12= 32 "red cards or face cards".

    Either way, the point is to be careful not to count "red face cards" twice.
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  4. #4
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    Re: Set Cardinalities

    Thank you so much!
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