# 0.10 Sets and counting

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This chapter covers principles of sets and counting. After completing this chapter students should be able to: use set theory and venn diagrams to solve counting problems; use the multiplication axiom to solve counting problems; use permutations to solve counting problems; use combinations to solve counting problems; and use the binomial theorem to expand x+y^n.

## 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.

$C=\left\{\text{Ken, Bob, Tran, Shanti, Eric}\right\}$

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

Let $A$ and $B$ be two sets, then the union of $A$ and $B$ , written as $A\cup B$ , is the set of all elements that are either in $A$ or in $B$ , or in both $A$ and $B$ .

## Intersection of two sets

Let $A$ and $B$ be two sets, then the intersection of $A$ and $B$ , written as $A\cap B$ , is the set of all elements that are common to both sets $A$ and $B$ .

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

## Complement of a set

Let $A$ be any set, then the complement of set $A$ , written as $\stackrel{ˉ}{A}$ , is the set consisting of elements in the universal set $U$ that are not in $A$ .

## Disjoint sets

Two sets $A$ and $B$ are called disjoint sets if their intersection is an empty set.

List all the subsets of the set of primary colors $\left\{\text{red, yellow, blue}\right\}$ .

The subsets are ∅, $\left\{\text{red}\right\}$ , $\left\{\text{yellow}\right\}$ , $\left\{\text{blue}\right\}$ , $\left\{\text{red, yellow}\right\}$ , $\left\{\text{red, blue}\right\}$ , $\left\{\text{yellow, blue}\right\}$ , $\left\{\text{red, yellow, blue}\right\}$

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

Let $F=\left\{\text{Aikman, Jackson, Rice, Sanders, Young}\right\}$ , and $B=\left\{\text{Griffey, Jackson, Sanders, Thomas}\right\}$ . Find the intersection of the sets $F$ and $B$ .

The intersection of the two sets is the set whose elements belong to both sets. Therefore,

$F\cap B=\left\{\text{Jackson, Sanders}\right\}$

Find the union of the sets $F$ and $B$ given as follows.

$F=\left\{\text{Aikman, Jackson, Rice, Sanders, Young}\right\}B=\left\{\text{Griffey, Jackson, Sanders, Thomas}\right\}$

The union of two sets is the set whose elements are either in $A$ or in $B$ or in both $A$ and $B$ . Therefore

$F\cup B=\left\{\text{Aikman, Griffey, Jackson, Rice, Sanders, Thomas, Young}\right\}$

Observe that when writing the union of two sets, the repetitions are avoided.

Let the universal set $U=\left\{\text{red, orange, yellow, green, blue, indigo, violet}\right\}$ , and $P=\left\{\text{red, yellow, blue}\right\}$ . Find the complement of $P$ .

The complement of a set $P$ is the set consisting of elements in the universal set $U$ that are not in $P$ . Therefore,

$\stackrel{ˉ}{P}=\left\{\text{orange, green, indigo, violet}\right\}$

To achieve a better understanding, let us suppose that the universal set $U$ represents the colors of the spectrum, and $P$ the primary colors, then $\stackrel{ˉ}{P}$ represents those colors of the spectrum that are not primary colors.

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