The blue circle of the Venn diagram depicts the number of students who prefer TV, the orange circle depicts the number of students who prefer radio, and the region of overlap indicates the number of students who like both. Therefore, 7 students like both TV and radio.

Start by removing the number who don't like either from the total number of peole surveyed:

$\displaystyle 30-6=24$

Using set notation we get:

$\displaystyle n(B\bigcup P)=n(B)+n(P)-n(B\bigcap P)$, or the number of people who like both colors, $\displaystyle n(B\bigcup P)$, is equal to the number of people who like blue, $\displaystyle n(B)$, plus the number of people who like pink, $\displaystyle n(P)$, minus the number of people who like both blue and pink, $\displaystyle n(B\bigcap P)$.

$\displaystyle 24=18+21-x$

$\displaystyle 24=39-x$

$\displaystyle x=15$

Start by removing the number who don't like either from the total number of students surveyed:

$\displaystyle 25-5=20$

Using set notation we get:

$\displaystyle n(F\bigcup L)=n(F)+n(L)-n(F\bigcap L)$, or the number of people who like both pets, $\displaystyle n(F\bigcup L)$, is equal to the number of people who like frogs, $\displaystyle n(F)$, plus the number of people who like lizards, $\displaystyle n(L)$, minus the number of people who like both frogs and lizards, $\displaystyle n(F\bigcap L)$.

$\displaystyle 20=13+15-x$

$\displaystyle 20=28-x$

$\displaystyle x=8$

The middle portion of the diagram is the area that both circles share, so the color name that belongs in both circles should go in the middle area. Doug and Jill both like green and tan, so those colors should go in the middle. Only Jill likes blue and yellow, so these go on Jill's side. Only Doug likes red and black, so these go on Doug's side.

A Venn Diagram is made up of two circles.  Each circle represents something, and the section where they intersect shows what those two circles have in common.  So, in the Venn Diagram

we can see Andy's hobbies on the left and Mary's hobbies on the right.  The section in the middle shows the hobbies Andy and Mary have in common.  

So, to answer the question, what hobbies do Andy and Mary have in common, we can see that they both enjoy eating and singing.

Let's look at the Venn Diagram.

We can see the first circle contains Andy's hobbies, and the second circle contains Mary's hobbies.  The place in the middle, where the 2 circles intersect, shows the hobbies that both Andy and Mary enjoy.  Or the hobbies they have in common.  You can see they are a part of each person's circle.

So, the hobbies that are in the middle are eating and singing.  Therefore, the hobbies that Andy and Mary have in common are eating and singing.

Start by removing the number who don't like either team from the total number of people surveyed:

$\displaystyle 50-10=40$

Using set notation we get:

$\displaystyle n(Chicago\bigcup St.Louis)=n(Chicago)+n(St.Louis)-n(Chicago\bigcap St.Louis)$, or the number of people who like both teams, $\displaystyle n(Chicago\bigcup St.Louis)$, is equal to the number of people who like the Cubs, $\displaystyle n(Chicago)$, plus the number of people who like the Cardinals, $\displaystyle n(St.Louis)$, minus the number of people who like both teams, $\displaystyle n(Chicago\bigcap St.Louis)$.

$\displaystyle 40=23+35-x$

$\displaystyle 40=58-x$

$\displaystyle x=18$

The grayed region is the set of all elements that are in $\displaystyle A \cap \overline{B }$ - that is, in $\displaystyle A$ but not $\displaystyle B$. This set is 

$\displaystyle A \cap \overline{B } = \left \{ 1,5,17,20\right \}$