0.10 Sets and counting  (Page 2/9)

We do the problems in steps.

We will use Venn diagrams to solve this problem.

Let the set $A$ represent those car enthusiasts who drove cars with automatic transmissions, and set $S$ represent the car enthusiasts who drove the cars with standard transmissions. Now we use Venn diagrams to sort out the information given in this problem.

Since 12 people drove both cars, we place the number 12 in the region common to both sets.

Because 30 people drove cars with automatic transmissions, the circle A must contain 30 elements. This means $x+12=30$ , or $x=18$ . Similarly, since 20 people drove cars with standard transmissions, the circle B must contain 20 elements, or $y+12=20$ which in turn makes $y=8$ .

Now that all the information is sorted out, it is easy to read from the diagram that 18 people drove cars with automatic transmissions only, 12 people drove both types of cars, and 8 drove cars with standard transmissions only. Therefore, $18+12+8=38$ people took part in the survey.

The problem is similar to the one in [link] .

Let the set $D$ represent the people who have visited Disneyland, and $K$ the set of people who have visited Knott's Berry Farm.

We fill the three regions associated with the sets D and K in the same manner as before. Since 100 people participated in the survey, the rectangle representing the universal set U must contain 100 objects. Let x represent those people in the universal set that are neither in the set D nor in K. This means $54+6+9+x=100$ , or $x=31$ .

Therefore, there are 31 people in the survey who have visited neither place.

Let $J$ represent the set of people who jog, $S$ the set of people who swim, and $C$ who cycle.

In using Venn diagrams, our ultimate aim is to assign a number to each region. We always begin by first assigning the number to the innermost region and then working our way out.

We place a 3 in the innermost region of [link] because it represents the number of people who participate in all three activities. Next we compute x, y and z.

Since 14 people jog and swim, $x+3=14$ , or $x=11$ .

The fact that 9 people jog and cycle results in $y+3=9$ , or $y=6$ .

Since 7 people swim and cycle, $z+3=7$ , or $z=4$ .

This information is depicted in [link] .

Now we proceed to find the unknowns m, n and p.

Since 50 people jog, $m+11+6+3=50$ , or $m=30$ .

Thirty people swim, therefore, $n+11+4+3=30$ , or $n=12$ .

Thirty five people cycle, therefore, $p+6+4+3=35$ , or $p=22$ .

By adding all the entries in all three sets, we get a sum of 88. Since 100 people were surveyed, the number inside the universal set but outside of all three sets is 100 – 88, or 12.

In [link] , the information is sorted out, and the questions can readily be answered.

1. The number of people who jog but do not swim or cycle is 30.
2. The number who take part in only one of these activities is $30+12+22=64$ .
3. The number of people who do not take part in any of these activities is 12.