# Sets

Group problems regard populations of persons or objects and the way these populations are grouped together into categories. The questions generally ask you to either find the total of a series of groups or determine how many people or objects make up one of the subgroups.

You can find the answer to most group problems by using your counting skills, but counting is time-consuming, and you want to work smarter, not harder, to solve these questions. Solving group problems comes down to applying simple arithmetic in a handy formula and nothing else.

Here is the formula for solving group problems:

Group 1 + Group 2 - Both Groups + Neither Group = Grand Total

For example, if out of 110 students, 47 are enrolled in a cooking class, 56 take a welding course, and 33 take both cooking and welding, you can use the formula to find out how many students take neither cooking nor welding. Let Group 1 be the cooks and Group 2 the welders. The variable is the group that doesn’t take either the cooking or welding class. Plug the known values into the formula and set up an equation to solve:

Group 1 + Group 2 - Both Groups + Neither Group = Grand Total

47 + 56 - 33 + x = 110

x = 40

Of the 110 students, 40 take neither the cooking class nor the welding class.

### Set

Groups are related to sets. A set is a collection of objects, numbers, or values. The objects in a set are the elements or members of the set. An empty set, or null set, means that nothing is in that set.

The relationship between sets is illustrated with a Venn diagram.

If S is a set having a finite number of elements, then the number of elements is denoted by |S|. A set is often defined by listing its elements. For example, S = {-5,0,1} is a set with |S| = 3.

### Set Terminology

The terms union, intersection, disjoint sets, and subset describe how two or more sets relate to one another through the elements they contain.

1. A union of two sets contains the set of all elements of both sets. For example, the union of sets A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {2, 4, 6, 8, 10} is S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

2. An intersection of two sets is the set of the elements that are common to both sets. For example, the intersection of sets A = {0, 1, 2, 3, 5, 6, 7, 8, 9} and B = {2, 4, 6, 8, 10} is S = {2, 6, 8}.

3. Disjoint sets are two or more sets with no elements in common. For example, set A and set B are disjoint sets if set A = {0, 2, 6, 8} and set B = {1, 3, 5, 7}.

4. A subset is a set whose elements appear in another, larger set. If all the elements of set B = {2, 3, 5, 7} also appear in set A = {0, 1, 2, 3, 5, 6, 7, 8, 9}, then set B is a subset of set A.

### Two Sets

For any two sets A and B, the union of A and B is the set of all elements that are in A or in B or in both. The intersection of A and B is the set of all elements that are both in A and in B.

For 2 sets: A and B: P(A u B) = P(A) + P(B) - P(A n B)

When counting elements that are in overlapping sets, the total number is equal the number in one group plus the number in the other group minus the number common to both the groups.

### Three Sets

For 3 sets: A, B, and C: P(A u B u C) = P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + P(A n B n C)

### Venn Diagrams

Venn diagrams provide visual representations of union, intersection, disjoint sets, and subset. You can draw Venn diagrams to help you answer GMAT questions about sets. 