A gift store stocks baseball hats in red or green colors. Of the 35 hats on display on a given day, 20 are green. As well, 18 of the hats have a grasshopper logo on the brim. Suppose 11 of the red hats have logos. How many hats are red, or have logos, or both?

The answer in the book is 22. I do not understand how they got this. Could someone please help me? Thanks.

It is not clear if you have learned the inclusion/exclusion principle. If you have, you can apply it without drawing the Venn diagram.

However, for a two-set problem like this, it is easier to draw a Venn diagram and solve accordingly.

There are three sets in a universal set E where the cardinality (i.e. total number of elements) |E|=35.
We are also given that for the set of green hats G, |G|=20.

We conclude therefore that for the set R, |R|=|E|-|G|=35-20=15.

Of the 35 hats, irrespective of colour, 18 of them have logos, so belong to the set L, where |L|=18.

We are required to find the set of hats which are either red, or has a logo, that is, the cardinality of the set R∪L, or |R∪L|.

Consider now the sets L and R.
Draw a Venn diagram for the two, with an intersection, i.e. both red and have a logo. We understand that |L∩R|=11.
So if you put in the Venn diagram 18 for L, 15 for R, and 11 for R∩L. You can calculate that L-R (i.e. with a logo but not red) is 18-11=7, and R-L (i.e. red but no logo) is 15-11=4.

So therefore
|R∪L| = 7+11+4 = 22.

Thanks a lot.

You're welcome!

To solve this problem, we need to analyze the information provided step by step.

First, we are given that there are 35 hats on display, and out of these, 20 are green. Therefore, the number of red hats can be found by subtracting the number of green hats from the total number of hats:

Number of red hats = Total number of hats - Number of green hats
= 35 - 20
= 15

Next, we are told that 18 of the hats have a grasshopper logo on the brim. Out of these 18, 11 are red hats. Therefore, the number of hats with logos that are not red can be found by subtracting the number of red hats with logos from the total number of hats with logos:

Number of hats with logos that are not red = Total number of hats with logos - Number of red hats with logos
= 18 - 11
= 7

To find the total number of hats that are red, or have logos, or both, we need to add the following:

Number of red hats + Number of hats with logos that are not red
= 15 + 7
= 22

Therefore, the total number of hats that are red, or have logos, or both is 22.

I hope this explanation clarifies how the answer is obtained. Let me know if you have any further questions!