Two basic question on set theory

Question

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 ?

Answer 1

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 $a+x=63, b+x=39$

Answer 2

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

Answer 3

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.

Answer 4

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%.