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