Problem 27 - Algebra and Trigonometry

Question

In a consumer survey of 500 people, 200 indicated that they would be buying a major appliance within the next month, 150 indicated that they would buy a car, and 25 said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car?

Step-by-Step Solution

Answer

175 will purchase neither. 125 will purchase only a car.

Step 1: Define the Sets

Let’s define two sets: Let A be the set of people buying a major appliance, and B be the set of people buying a car. Given: \( |A| = 200 \)\( |B| = 150 \)\( |A \cap B| = 25 \)

Step 2: Calculate the Union of Sets A and B

To calculate how many people are buying either a major appliance or a car or both, use the formula: \( |A \cup B| = |A| + |B| - |A \cap B| \) Substituting the numbers: \( |A \cup B| = 200 + 150 - 25 = 325 \)

Step 3: Calculate the Number of People Who Will Purchase Neither

The total number of people surveyed is 500. To find the number of people who will purchase neither a major appliance nor a car, subtract the number of people purchasing either from the total: \( 500 - 325 = 175 \)

Step 4: Calculate the Number of People Who Will Purchase Only a Car

To find the number of people who will purchase only a car, subtract those who will buy both from the set of people buying a car: \( |B| - |A \cap B| = 150 - 25 = 125 \)

Key Concepts

Union of Sets

The union of two sets, denoted as \(A \cup B\), represents all elements that are in set A, or set B, or in both. In the context of our exercise, we could think of it as people buying either a major appliance or a car, or both. To find the union, we use a simple formula:

\( |A \cup B| = |A| + |B| - |A \cap B| \)
This formula helps avoid double-counting the elements that appear in both sets. Let's plug in the numbers from our exercise:

\( |A \cup B| = 200 + 150 - 25 = 325 \)
This tells us that 325 people are buying either a major appliance or a car, or both. Understanding the notion of union helps in various areas of math and logic.

Intersection of Sets

The intersection of two sets, denoted as \(A \cap B\), consists of elements that are common to both sets. For our exercise, this is the group of people who will buy both a major appliance and a car. Given:

\( |A \cap B| = 25 \)
It tells us that 25 people will be buying both a major appliance and a car.

Intersections help us understand how sets overlap. This concept is not just limited to consumer surveys, it is used in fields such as biology (think of species overlapping in habitats), database searches (finding common records), and even in daily activities (like overlapping friends attending parties).
Remember when you solve these problems to consider all overlapping elements carefully.

Complement of a Set

The complement of a set, often denoted as \(A'\) or \(B'\), refers to all elements not in that set. For our exercise, we're interested in people who will purchase neither a major appliance nor a car.

We already found that 325 people are buying either a major appliance or a car or both.
To get our complement, we subtract this union count from the total population (which is 500):

500 - 325 = 175
So 175 people will not buy a major appliance or a car. Understanding complements helps when you need to determine what is outside a given set, often useful in probability and logical condition validations.
For instance, if asked how many students didn't pass either of two exams given the numbers who passed at least one, similar principles can be applied.