Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers essential counting concepts including finding all subsets of a given set using an organized approach, understanding set relationships through subset, proper subset, and empty set definitions, and applying key counting principles such as the multiplication and addition principles. Mastery of these fundamentals not only supports solving abstract counting problems but also real-world scenarios like survey analysis and menu selection.

Learning Objectives

1

Identify all subsets of a set and organize them by the number of elements.

2

Define and use set notation such as subset (?), proper subset (?), and the empty set (?).

3

Apply the counting formula for finite sets, including the 2^n rule for subsets.

4

Utilize the addition and multiplication principles to solve real-world counting problems.

Key Concepts

CONCEPT

DEFINITION

Subset

A set A is a subset of B (A ⊆ B) if every element of A is also an element of B.

Proper Subset

A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B, meaning B has at least one element not in A.

Empty Set

The set with no elements, denoted by ∅, which is a subset of every set.

Counting Notation n(A)

Represents the number of elements in set A.

Multiplication Principle

If a task involves a sequence of choices with p, q, r, ... options respectively, then the total number of ways to complete the task is p × q × r × ... .

Addition Principle

When two sets are disjoint, the number of elements in their union is the sum of the elements in each set; for overlapping sets, subtract the common elements to avoid double counting.

Counting Formula for Subsets

For a finite set with n elements, the total number of subsets is 2^n.

Example Problems

Example 1

The________________ of $A$ and $B$ consists of all elements in either $A \text { or } \bar{B} \text { or both. (pp. } 2-3)$

Example 2

The __________ of $A$ with $B$ consists of all elements in both $A$ and $B .(\mathrm{pp} .2-3)$

Example 3

True or False The intersection of two sets is always a subset of their union. (pp. $2-3)$

Example 4

True or False If $A$ is a set, the complement of $A$ is the set of all the elements in the universal set that are not in $A$. $(\mathrm{pp} .2-3)$

Example 5

If each clement of a set $A$ is also an element of a set $B,$ we say that $A$ is____________ a $\quad$ of $B$ and write $A$___________ $B$ .
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Step-by-Step Explanations

QUESTION

List all subsets of the set {a, b, c}.

STEP-BY-STEP ANSWER:

Step 1: Write the subset with 0 elements: ∅.
Step 2: Write all subsets with 1 element: {a}, {b}, {c}.
Step 3: Write all subsets with 2 elements: {a, b}, {a, c}, {b, c}.
Step 4: Write the subset with 3 elements: {a, b, c}.
Final Answer: The subsets of {a, b, c} are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}.

Finding All Subsets

QUESTION

How many different meals can be ordered at a restaurant offering 2 appetizers, 4 entrées, and 2 desserts?

STEP-BY-STEP ANSWER:

Step 1: Identify the number of choices for each course: 2 choices for an appetizer.
Step 2: There are 4 choices for an entrée.
Step 3: There are 2 choices for a dessert.
Step 4: Multiply the number of choices: 2 × 4 × 2 = 16.
Final Answer: There are 16 different meals possible.

Multiplication Principle

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Common Mistakes

  • Confusing a subset (?) with a proper subset (?), especially neglecting that equality is allowed in subsets.
  • Forgetting to include the empty set as a valid subset of any set.
  • Double counting elements when applying the addition principle for overlapping sets.
  • Misapplying the multiplication principle by not properly identifying independent choices in a sequence.