1. ## 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.

2. ## 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

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

3. ## 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.

4. ## Re: Set Cardinalities

Thank you so much!