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I have run into this question on my practice exam, I'm not quite sure how to approach it. Given $|U|=80,|A|=30,|B|=42, |A ∩ B| = 9$, I'm supposed to determine $|A^c ∩ B^c|$

I understand the meanings of all the symbols, cardinality being how many elements are in the set, ∩ being intersection, and $^c$ being whatever is not in the set, but I still don't have a clue as to how that relates to how many are in the intersection of the elements not in their sets, please help!

Thanks in advance!

Scientifica
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ay lmao
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    $A^c\cap B^c = (A\cup B)^c$ and $|A\cup B| = |A| + |B| - |A\cap B|$ – Prahlad Vaidyanathan Mar 13 '15 at 02:14
  • @PrahladVaidyanathan I don't know why other answers are being given here. Your comment was more than enough to actually be an answer IMO. The only thing additional that may be helpful to add is that this is simply a particular case of using inclusion-exclusion. – Daniel W. Farlow Mar 13 '15 at 02:19
  • @crash I don't know why you think the initial comment is in any way explanatory. It lists an identity, and doesn't explain the process to arrive at this identity. Not everyone can be as smart as you. – layman Mar 13 '15 at 02:22
  • @user46944 Firstly, it has nothing at all to do with "how smart I am." I'm not that smart. Secondly, the identities (plural) mentioned in the comment are pretty standard. So if OP is being asked to calculate $|A^c\cap B^c|$, then s/he should, in all likeliness, have come across those identities. – Daniel W. Farlow Mar 13 '15 at 02:24
  • @crash If the OP understood those standard identities, I highly doubt they would be asking this question. In this case, listing the identities that the OP in all probability doesn't understand doesn't help them understand. That's why other answers are being given here. – layman Mar 13 '15 at 02:26
  • @user46944 How is Graham Kemp's answer at all useful or, in fact, much different than the comment? Because the names of the identities are given? Naming them was done in a subsequent edit, and it still doesn't add much help I don't think. So the real question then...OP: have you heard of deMorgan or inclusion-exclusion? (Using inclusive or here) – Daniel W. Farlow Mar 13 '15 at 02:29
  • @crash I'm not interested in justifying Graham Kemp's answer. I was defending my answer. – layman Mar 13 '15 at 02:30
  • @user46944 Ha--that would explain it; I didn't even notice you were the other person who answered. In that case, I would edit your answer to make it more generalized so that OP sees how the identities come in hand (or refer him/her on to links that explain those identities). That's all that's needed here obviously. Your way is fine, but it's pretty limited in terms of its usefulness IMO. – Daniel W. Farlow Mar 13 '15 at 02:32
  • @crash I don't think it's required to add all of that extra information just to solve the original problem the OP presented. But, if you see the usefulness, feel free to write an answer adding all of that information. I'm sure people will upvote. – layman Mar 13 '15 at 02:35
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    @user46944 It certainly isn't required at all, but sometimes a tad bit more effort can really help the OP. For example, check out this answer I gave the other day. Talk about so much more than was really required. But it should help OP and others in the future. Anyway, I don't care about upvotes, but I try to make an answer as helpful as possible. What you have done is good, but sometimes it helps to add a touch of flavor too :) – Daniel W. Farlow Mar 13 '15 at 02:39
  • @crash I agree with you. But I don't really know how to add the details you suggested in a way that won't confuse the OP based on the level I perceive them to be at. Maybe now that they understand the solution to this problem, it will help them understand the laws listed in future problems. – layman Mar 13 '15 at 02:51
  • @user46944 Could never hurt to add details as, for example, Edit: The reason for the comments...here are links...etc. Doesn't really matter now though, seeing as the OP is satisfied with your answer. – Daniel W. Farlow Mar 13 '15 at 02:57

2 Answers2

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You are essentially being asked to find how many elements are both not in $A$ and not in $B$.

Let's see how to work this out:

$U$ is our universal set, i.e., all elements are located in $U$ and $A$ and $B$ are subsets of $U$. We have $|U| = 80$, i.e., we have $80$ elements total. Since $|A| = 30$, $A$ has $30$ elements. Since $|B| = 42$, $B$ has $42$ elements. Since $|A \cap B| = 9$, that means $A$ and $B$ have $9$ elements in common.

If $A$ and $B$ have $9$ elements in common, and $A$ has $30$ elements total, that means $A$ has $30 - 9 = 21$ elements in it that are not in $B$. Similarly, since $B$ has $42$ elements, $B$ has $42 - 9 = 33$ elements in it that are not in $A$.

So, the elements that are either in $A$ or $B$ are those that are in $A$ and not in $B$, those in $B$ but not in $A$, and those in both $A$ and $B$. That means $A \cup B$ has $21 + 33 + 9 = 63$ elements. These are the elements that are either in $A$ or $B$.

That means elements that are neither in $A$ or nor $B$ are $80 - 63 = 17$ elements. That's how many are in $A^{c} \cap B^{c}$.

layman
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Hints:

$ \text{de Morgan's rule: }X^\complement \cap Y^\complement = (X\cup Y)^\complement \\ \text{Rule of Complements: }\lvert X^\complement \rvert = \lvert U\rvert - \lvert X\rvert \\ \text{Rule of Total Measure: } \lvert X\cup Y\rvert = \lvert X\rvert + \lvert Y\rvert - \lvert X\cap Y\rvert $

Use them.

Graham Kemp
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