Overlapping Sets

Question 1

Overlapping Set Problems:

Answer 1

If a question gives you a lot of data about different sets that partially intersect with each other and asks you for a specific missing number, see if you can organize the data into a chart. That makes it easy to calculate the missing data.

In an overlapping sets problem if a question stem does not have any natural numbers or percentages at all work with variables (A, B, C, C) and use the information given to set up expressions with the variables (e.g. question stem says house y is three times bigger than house x then y = 3x)

Question 2

Overlapping Sets:

Answer 2

There are 3 approaches to solving overlapping sets problems where you are presented with complex relationships between groups and subgroups. Practice all three to see which one YOU like best with which problems.

1. OVERLAPPING SETS FORMULA:

Group 1 + Group 2 - Both + Neither = Total

This approach is good if the question stem offers numbers for the “both” and “neither” categories or gives you information on them.

E.g. Someone orders 27 pizzas. 15 have pepperoni, 10 have mushrooms. If 4 pizzas have no toppings at all, and no other toppings were ordered, how many pizzas were ordered with both pepperoni and mushrooms?

15 + 10 - Both +4 = 27
                    Both = 2

2. VENN DIAGRAM which uses partially overlapping circles to present the data visually. It’s used in three-set problems. Where the circles overlap are the subgroups of both or all circles. Each circle minus the overlapping area represents the group that’s only that group and not the group of the other circle(s). The total number is the sum of one circle minus the overlapping area and the other circle(s) minus the overlapping area(s).
3. CHART (double-set matrix) is good for complicated overlapping sets problems because it has a separate place for each of the nine data points you might be given or have to find in order to get to the solution. Organizing the chart is most of the work, the results then only take simple calculations.

Question 3

Double-set matrix (Chart):

Answer 3

For problems where you have only two categorizations or decisions, i.e. problems with only two sets of options, use the double-set matrix. In contrast to Venn diagram it displays all combinations of options, including totals.

Often times the sets consist of four categories similar to these:

1. Things that belong to A
2. Things that belong to B
3. Things that belong to both A and B
4. Things that belong to neither A nor B

Each row and each column sum to a total.

When you set up the matrix be careful that you don’t mix up mutually exclusive options. For instance there shouldn’t be a box that both leads to “A” and “Not A” because either it is A or it isn’t. It can’t be both.

REMEMBER:

• For problems involving percents pick 100 for total
• For problems involving fractions, pick common denominator for total

If you don’t have time to complete the entire matrix, only calculate as far as you have to in order to get the answer.

Question 4

Double-Set Matrix and Three Choices:

Answer 4

In some questions (rare) you have to consider more than two options for one or both dimensions of the double-set matrix. As long as each option is distinct you can just add a choice to the matrix. E.g. in survey respondents could answer “Yes,” “No,” or “Maybe” and there were men and women in the survey. so you set up a matrix with three columns at top (yes, no, maybe) and 2 rows on side (men, women).

You rarely have to do the math for these type of questions. But for some DS questions setting up a multiple-choice matrix like that can be helpful.

Question 5

Venn Diagram:

Answer 5

Venn Diagrams are used in three-set problems, that is problems that involve three overlapping sets.

REMEMBER: When the options don’t overlap use double-set matrix. When they do overlap use Venn Diagram.

Three overlapping sets in Venn Diagram are usually three teams, clubs. Each person is either on or not on any given team. I.e. there are only two choices for any team: member or not.

Venn Diagram is three circles that overlap and create 7 sections total. You’ll have one section in the middle, the innermost section, in which all three circles overlap and three other sections around that section where two circles each overlap. Plus, you have three outermost sections.

REMEMBER: Important rule for working with Venn Diagrams: Work from inside out.

First put in value for innermost section (where all three circles overlap). Then add the values of the three sections surrounding the innermost. Here you have to remember: those are people/entities that are on two teams. That means you must remember to subtract from that number of members of two teams the members of all three teams, i.e. number of innermost section. You have to do that for all three surrounding sections. Then you go on to the outermost sections. Here you will have to subtract from the total number you were given for each team, the overlapping sections respectively. To find out the total number of members (of all three teams/clubs/circles combined) add all 7 section numbers together.

REMEMBER: If you are asked how many people are in AT LEAST one of the groups, you have to just sum all numbers in the Venn Diagram, from the outermost to the innermost sections.