The grayed region is the set of all elements that are in  - that is, in  but not . This set is 

The grayed region is the set of all elements that are in  - that is, in both  and . This set is 

A Venn diagram is used to demonstrate the number of people (or things) in particular categories. The regions of the circle that are overlapping indicate the number of individuals that belong to multiple groups. Therefore, based on this diagram, 12 students like soccer only, 16 like football only, and 8 like both.

The shaded area where the circle that contains the word guitar overlaps with the circle that contains the word drums has the number 5.  It is incorrect to add the total number of guitar players to the total number of drum players because we are only looking for students who play both instruments.  

18 students play the guitar, but you must also add the 5 students who play the guitar and drums and the 3 students who play the guitar and piano.  

There are 60 total students. If 1 person plays all 3 instruments, then add that to all of the other numbers on the Venn Diagram to see that  58 students are represented on the Venn Diagram. Subtract that number from the total number of students to find the number of students in the missing category (those who play both the drums and the piano, but not the guitar).

The intersection of a venn diagram represents when both events happen at the same time, not when neither happen.

When looking at a Venn Diagram, we can see that it is made up of two circles.  In this Venn Diagram

we can see Andy's hobbies on the left and Mary's hobbies on the right.  We also see what hobbies they have in common in the center where the two circles intersect.  

Now, when looking at Andy's circle containing his hobbies, we see that he enjoys sleeping, reading, and cooking.  His circle is also part of the intersection, so we must include those hobbies as well.  So, he also enjoys eating and singing. 

So, overall, Andy's hobbies include sleeping, reading, cooking, eating, and singing.

The point where two circles on a Venn Diagram intersect is the point where both of the events occur, not when neither occur.

When looking at a Venn Diagram, we can see that it is made up of two circles.  In this Venn Diagram we can see Andy's hobbies on the left and Mary's hobbies on the right.  We also see what hobbies they have in common in the center where the two circles intersect.  

Now, when looking at Andy's circle containing his hobbies, we see that he enjoys sleeping, reading, and cooking.  His circle is also part of the intersection, so we must include those hobbies as well.  So, he also enjoys eating and singing. 

So, overall, Andy's hobbies include sleeping, reading, cooking, eating, and singing.