Problem 23 - Discrete Mathematics with Applications

Question

A survey was taken to determine the preference between two laundry detergents, Lex and Rex. It was found that 15 people liked Lex only, 10 liked both, 20 liked Rex only, and 5 liked neither of them. How many people were surveyed?

Step-by-Step Solution

Answer

The total number of people surveyed can be calculated as follows: \(15 + 10 + 20 + 5 = 50\). Therefore, 50 people were surveyed in total.

Step 1: Identify given information

From the problem, we are given the following information: - 15 people liked Lex only - 10 people liked both Lex and Rex - 20 people liked Rex only - 5 people liked neither Lex nor Rex

Step 2: Sum up the number of people in each group

Add the number of people in each group to find the total number of people surveyed. Total number of people surveyed = Number of people who liked Lex only + Number of people who liked both Lex and Rex + Number of people who liked Rex only + Number of people who liked neither Lex nor Rex Total number of people surveyed = 15 + 10 + 20 + 5

Step 3: Calculate the total number of people surveyed

Perform the addition: Total number of people surveyed = 15 + 10 + 20 + 5 = 50 There were 50 people surveyed in total.

Key Concepts

Discrete Mathematics

Discrete mathematics encompasses a variety of topics that are fundamental to computer science, economics, and the sciences. When it comes to set theory—a key part of discrete mathematics—it involves the study of well-defined collections of objects, known as sets. These can be anything from numbers to letters, or even theoretical concepts. The study of sets is crucial for understanding mathematical concepts and solving problems like the laundry detergent survey. In this case, counting the number of people that fall into different groups requires discrete mathematics because the individuals surveyed can be clearly separated into distinct categories or sets based on their preferences.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a critical concept in set theory that prevents double-counting when determining the size of the union of two or more sets. When we say someone likes 'both' detergents in the survey problem, we need to ensure we're not overcounting these individuals in the total. This principle allows us to figure out the exact number of unique survey responses by adding the number of people for each preference and then subtracting the number counted twice for liking both brands. It's a fundamental rule for navigating complex set-related problems that ensure our computations reflect the true distribution of elements across various categories.

Venn Diagrams

Venn diagrams are an incredibly intuitive tool used in set theory to visually depict the relationships between different sets. By drawing circles that represent different sets and their overlapping areas, we can easily see the number of elements that belong exclusively to one set, those that belong to intersecting sets, and those that do not belong to any set. A Venn diagram would enhance the understanding of the laundry detergent problem by clearly showing the overlap between people who like Lex, those who like Rex, and those who like both or neither. Using Venn diagrams simplifies complex set relationships and aids in applying the inclusion-exclusion principle.

Set Operations

The various operations you can perform on sets, such as union, intersection, and complement, are similar to arithmetic for numbers but are used to combine or compare groups of elements. The union of sets combines all unique elements, the intersection finds those common to all, and the complement contains those not in the set. For the detergent survey, understanding that the 'both' category needs to be added in full but only counted once is an example of applying set operations to avoid miscalculations in the total. Mastery of these operations is essential for accurately tabulating and analyzing data from surveys and studies.