Sets - some tricky problems!!

Hi \'manpreet singh:

Q.1) There are 6 stores in town that had a total of 20 visitors on a particular day.However, only 10 people went shopping that day; some people visited more than one store. If 6 people visited exactly two stores each, and everyone visited at least one store, what is the largest number of stores anyone could have visited?

This is how I would do them. Start with the maximum and keep taking out the numbers on both sides.
First I draw a grid like this. We have 10 people and 20 visits overall

I would draw it like this
20-------10

Then 6 people visited exactly two stores each, so a total of 12 stores.
So we have 12----------6.

I would then subtract them both.

20----------10
12----------6
8------------4

At this point I know that I have 4 people with 8 visits. Only one needs to visit the maximum number and the rest need to visit at least 1 store. So I have to minimize 3, and the rest will be maximized. So for those 3, we have:
3-----------3

Subtracting this from the last one

8----------4
3----------3
5----------1
We get 1 person can visit maximum of 5 stores. This is how I would do them. In reality, you can just keep doing this process like a continued subtraction, as you keep reading the conditions.

Q.2) There are 26 students who have read a total of 56 books among them. The only books they have read, though, are Aye, Bee, Cod, and Dee. If 10 students have only read Aye, and 8 students have read only Cod and Dee, what is the smallest number of books any of the remaining students could have read?

Again, going the same way that I came up with in the earlier post, we have 56 books and 26 people. So we start with

56------------26

Now we know that 10 people read only Aye, so

56------------26
10-------------10
46-------------16

So at this point we have 16 people and 46 books read. Also, 8 people read only Cod and Dee. So 16 books are read by those people. Then, subtracting in the same way.

46--------------16
16--------------8
30--------------8.

So at this point I have 8 people left and 30 book readings left. Only 1 needs to read the minimum number. To minimize the 1, I will maximize the other 7. The maximum number of books anyone can read is 4 (since there are only 4 books). So total readings by these 7 = 7*4= 28.

So we continue with this. Subtracting the bottom one which we just thought of from the top one, we get
30---------------8
28---------------7
2-----------------1

So one person can read a minimum of 2 books.

Q.3) X and Yare sets of integers. X | Y denotes the set of integers that belong to set X or set Y,but not both. If X consists of 10 integers, Y consists of 18 integers, and 5 of the integers are in both X and Y,then X | Y consists of how many integers?

In this one we are asked for an overlapping sets problem where we need to find the integers unique to each set.
This is how I think about it.

Items unique to X = Total Items in X – Total Items in XY. = X – XY
The same way, Items Unique to Y = Y-XY.

So X|Y = X-XY + Y-XY = 10+18 – 2*5 = 18.

I don’t know whether there is a better or more general way of doing the first two, but this is what I thought of when I read the questions.

Let me know if this helps

There are 6 stores in town that had a total of 20 visitors on a particular day.However, only 10 people went shopping that day; some people visited more than one store. If 6 people visited exactly two stores each, and everyone visited at least one store, what is the largest number of stores anyone could have visited?

Since 6 people visited exactly two stores each, the total number of visits attributed to these 6 people = 6*2 = 12.

Since there are a total of 20 visits, the number of visits remaining = 20-12 = 8.
Since there are a total of 10 shoppers, the number of shoppers remaining = 10-6 = 4.

To MAXIMIZE the number of stores visited by one of these 4 people, we must MINIMIZE the number of stores visited by the other 3 people.
Since each of the other 3 people must visit at least 1 store, the minimum number of visits that could be delegated to these 3 people = 3*1 = 3.

The number of remaining visits = 8-3 = 5.
Since the one remaining person could use these 5 visits to shop at 5 different stores, the largest number of stores that anyone could have visited = 5.

'manpreet singh wrote: X and Y are sets of integers. X | Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 5 of the integers are in both X and Y, then X | Y consists of how many integers?

Draw a Venn diagram and work from the inside out:


Step 1: Fill in the OVERLAP
Since 5 integers are in BOTH sets, put 5 in the OVERLAP between the two circles.

Step 2: SUBTRACT the overlap
Since X=10, only X = 10-5 = 5.
Since Y=18, only Y = 18-5 = 13.

X | Y = only X + only Y = 5+13 = 18.

Thanks Mitch and eagle, it was helpful!!! i like the way you guys approach the problem and make it look simple..