GMAT Quantitative Review

3.0 Math Review

3.1 Arithmetic

9. Sets

In mathematics a set is a collection of numbers or other objects. The objects are called the elements of the set. If S is a set having a finite number of elements, then the number of elements is denoted by image. Such a set is often defined by listing its elements; for example, image is a set with image. The order in which the elements are listed in a set does not matter; thus image. If all the elements of a set S are also elements of a set T, then S is a subset of T; for example, image is a subset of image.

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. The union is denoted by image and the intersection is denoted by image. As an example, if image and image, then image and image. Two sets that have no elements in common are said to be disjoint or mutually exclusive.

The relationship between sets is often illustrated with a Venn diagram in which sets are represented by regions in a plane. For two sets S and T that are not disjoint and neither is a subset of the other, the intersection image is represented by the shaded region of the diagram below.

image

This diagram illustrates a fact about any two finite sets S and T: the number of elements in their union equals the sum of their individual numbers of elements minus the number of elements in their intersection (because the latter are counted twice in the sum); more concisely,

image.

This counting method is called the general addition rule for two sets. As a special case, if S and T are disjoint, then

image

since image.