Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on various counting methods including determining the number of subsets using 2^n, and solving problems using addition and multiplication principles. The inclusion-exclusion principle is emphasized to avoid double counting when sets have overlapping elements. Real-world applications such as analyzing survey responses, meal combinations, and code formation illustrate how these mathematical principles facilitate systematic counting and problem-solving.

Learning Objectives

1

Understand how to count the number of subsets of a set and apply the formula 2^n.

2

Explain and apply the Inclusion-Exclusion Principle (Counting Formula) for the union of two finite sets.

3

Utilize the Addition Principle when sets are disjoint and the Multiplication Principle for sequential choices.

4

Analyze real-world data using counting principles through survey data and menu selection problems.

Key Concepts

CONCEPT

DEFINITION

Subset

A set where every element is also an element of another set. The total number of subsets of a set with n elements is 2^n.

Counting Formula (Inclusion-Exclusion Principle)

For two sets A and B, the number of elements in their union is given by n(A ∪ B) = n(A) + n(B) - n(A ∩ B), which corrects for double counting.

Addition Principle of Counting

If two sets have no elements in common (i.e., are disjoint), the total number of outcomes is the sum of the outcomes for each set.

Multiplication Principle of Counting

When a task involves a sequence of independent choices, the total number of outcomes is the product of the number of choices at each step.

Universal Set

The complete set of all elements under consideration in a particular discussion or problem.

Example Problems

Example 1

The ____ of $A$ and $B$ consists of all elements in either $A$ or $B$ or both.

Example 2

The ___ of $A$ with $B$ consists of all elements in both $A$ and $B .

Example 3

The intersection of two sets is always a subset of their union.

Example 4

If $A$ is a set, the complement of $A$ is the set of all the elements in the universal set that are not in $A$.

Example 5

If each element of a set $A$ is also an element of a set $B,$ we say that $A$ is a ___ of $B$ and write $A$ ___$B$.
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Step-by-Step Explanations

QUESTION

How many subsets does the set {a, b, c} have?

STEP-BY-STEP ANSWER:

Step 1: Recognize that a set with n elements has 2^n subsets.
Step 2: Identify that n = 3 for the set {a, b, c}.
Step 3: Calculate 2^3 = 8.
Final Answer: The set {a, b, c} has 8 subsets.

Counting Subsets

QUESTION

In a survey, 35 students are registered in College Algebra, 52 in Computer Science I, with 18 in both. How many students are registered in at least one of the courses?

STEP-BY-STEP ANSWER:

Step 1: Write down the Inclusion-Exclusion Principle formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
Step 2: Substitute the values: 35 + 52 - 18.
Step 3: Compute the sum: 35 + 52 = 87.
Step 4: Subtract the intersection: 87 - 18 = 69.
Final Answer: 69 students are registered in at least one of the courses.

Union of Two Sets Using the Counting Formula

QUESTION

How many different meals can be ordered given 2 appetizer choices, 4 entrée choices, and 2 dessert choices?

STEP-BY-STEP ANSWER:

Step 1: Identify each decision point: appetizer, entrée, and dessert.
Step 2: Count the number of options for each: 2 appetizers, 4 entrées, and 2 desserts.
Step 3: Multiply the choices: 2 x 4 x 2.
Step 4: Compute the product: 16.
Final Answer: There are 16 different meals that can be ordered.

Counting Possible Meals Using the Multiplication Principle

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Common Mistakes

  • Failing to subtract the intersection when applying the Inclusion-Exclusion Principle, leading to double counting.
  • Confusing the union of sets with the intersection, especially in survey analysis problems.
  • Overlooking the order of choices in the multiplication principle, which can alter the total count of outcomes.
  • Misinterpreting scenarios where sets are not disjoint and incorrectly applying the Addition Principle without adjustments.