0.10 Sets and counting

Chapter overview

In this chapter, you will learn to:

1. Use set theory and Venn diagrams to solve counting problems.
2. Use the Multiplication Axiom to solve counting problems.
3. Use Permutations to solve counting problems.
4. Use Combinations to solve counting problems.
5. Use the Binomial Theorem to expand ${\left(x+y\right)}^n$ .

Sets

In this section, we will familiarize ourselves with set operations and notations, so that we can apply these concepts to both counting and probability problems. We begin by defining some terms.

A set is a collection of objects, and its members are called the elements of the set. We name the set by using capital letters, and enclose its members in braces. Suppose we need to list the members of the chess club. We use the following set notation.

A set that has no members is called an empty set . The empty set is denoted by the symbol Ø.

Two sets are equal if they have the same elements.

A set $A$ is a subset of a set $B$ if every member of $A$ is also a member of $B$ .

Suppose $C=\left\{\text{Al, Bob, Chris, David, Ed}\right\}$ and $A=\left\{\text{Bob, David}\right\}$ . Then $A$ is a subset of $C$ , written as $A\subseteq C$ .

Every set is a subset of itself, and the empty set is a subset of every set.

Union of two sets

Intersection of two sets

A universal set $U$ is the set consisting of all elements under consideration.

Complement of a set

Disjoint sets