# Two basic question on set theory

$$1$$.How many proper subset of $$\{1,2,3,4,5,6,7\}$$ contain the numbers $$1$$ and $$7$$ ?

Lets consider $$\{1,7\}$$ as a single element then the number of possible subset is $$2^6$$ and hence the number of proper subset is $$62$$.

$$2$$. A survey show that $$63$$% of the Americans like cheese where as $$73$$% like apples.If $$x$$% of Americans like both cheese and apples, then we have :

(A) $$x \ge 39$$ (B) $$x \le 63$$ (C) $$39 \le x \le 63$$ (D) None of these

if $$a$$% and $$b$$% like only cheese and only apples then we have, $$a + x + x + b = 100$$ , $$a + x = 63$$ and $$b + x = 63$$ solving we get $$x = 39%$$. So (D) is my answer.

Am I correct?

• I think you probably meant $b+x=39$. Oct 28, 2010 at 5:50
• Using your own logic, how can you get 62 proper subsets out 2^6 total subsets? Why are you subtracting two and not one? Oct 28, 2010 at 6:24

1. No, the subsets must have 1 and 7. The other five elements are optional, but you can't have them all.

2. You have $a + b + x + \text{those who like neither cheese nor apples} =100%$. Don't count x twice when adding to 100%. But you are right that $a+x=63, b+x=39$

• (1)I don't understand what you mean & for (2) now I have no idea how to solve it ?!
– Damir Frezivioski
Oct 27, 2010 at 23:52
• What!? Who couldn't like either cheese or apples? Oct 27, 2010 at 23:54
• @damir Number 1 is $2^5\space=\space32$ Oct 28, 2010 at 2:55
• @muntoo not quite. It calls for proper subsets. Oct 28, 2010 at 3:31
• @damir so the legal subsets are {1,7} plus all subsets of {2,3,4,5,6} except the one that is all of them. That would not give a proper subset. As there are five elements, there are 2^5-1=31 possibilities. The empty set is a proper subset. Oct 28, 2010 at 3:32

On #1, you are over counting. You have thought of {1,7} as a single elements but your answer has included the possibility that it is not in there. Think of it as we are setting 1 and 7 to the side. Now for each remaining element, we can either include it or exclude it from a subset. If you still don't see how to obtain the answer, I would recommend the following short explanation of the multiplication principle

On #2, you are correct in thinking that each person must fall into 1 of four groups: people who like apples and cheese, people who just like cheese, people who like just apples, and people who like neither. If you think abut the information you are given, you should see an upper bound for x. For a lower bound on x, assume that everyone either likes apple or cheese and use the inclusion exclusion principle

• In (1) I really don't understand why and how you are using multiplication principle. In (2) The lower bound determination is quite understandable but I am facing some troubles to understand how are you getting the upper bound ...:) Oct 28, 2010 at 5:45
• For the upper bound, we know that 63% of people like cheese and 73% of people like apples. The 'worst case' is that everyone who likes cheese also likes apples. Hence, at most 63% could like both. Oct 28, 2010 at 5:50
• I meant multiplication principle in the sense that the for each of the 5 sets they can be either in or out (2 options per element) hence we have 2^5 total, but we require proper subsets so subtract 1 off. Is this more clear? Oct 28, 2010 at 5:51
• +1, Aha ! so it's just a different way to look at the problem :) Oct 28, 2010 at 5:55

1. Say {1,7} is a single element, then total number of subsets possible = $2^6$ Now, how many subsets among these don't have {1,7} ? It will be $2^5$.

Hence,number of subsets having {1,7} is $2^6$ - $2^5$ = 32.But since you have asked for proper subset so the answer would be 31.

• You could also get there directly, by asking how many subsets you can make out of {2,3,4,5,6} which 2^5 then subtract the one which is the whole set and get 2^5 - 1 = 31 Oct 28, 2010 at 5:58
• Hm yes,but I showed him that way to match his way of thinking :) Oct 28, 2010 at 6:00
1. crasic has taken the words out of my mouth. It's the same as the number of proper subsets of {2, 3, 4, 5, 6}, i.e. 2^5 - 1 = 31.

2. Only 63% like cheese, so there cannot be more than 63% who like both cheese and apples. So B is correct.

37% don't like cheese, and 27% don't like apples (which I personally find hard to imagine). The lower bound for x is found in the case where these two sets are disjoint - 64% either don't like cheese or don't like apples, leaving 36%.