Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S=\\{1,2,3\\}\) be a sample space associated with an experiment. a. List all events of this experiment. b. How many subsets of \(S\) contain the number 3 ? c. How many subsets of \(S\) contain either the number 2 or the number 3 ?

Step-by-Step Solution

Answer

a. The events of this experiment are the subsets of the sample space S: {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}. b. There are 4 subsets containing the number 3: {3}, {1, 3}, {2, 3}, {1, 2, 3}. c. There are 6 subsets containing either the number 2 or the number 3: {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.

Step 1: List all events of the experiment

The events of this experiment are the subsets of the sample space S. To list all possible subsets, consider each element and whether it is included or not in the subset. Since there are 3 elements, each having 2 possibilities (in or out), there are a total of \(2^3 = 8\) subsets (including the empty set and the set S itself). The subsets are: 1. {} 2. {1} 3. {2} 4. {3} 5. {1, 2} 6. {1, 3} 7. {2, 3} 8. {1, 2, 3} These 8 subsets are the events of this experiment.

Step 2: Count the subsets containing the number 3

To count the subsets that contain the number 3, observe that now we only need to consider the elements 1 and 2 and whether they are included in the subset or not. For example, the subsets containing the number 3 could have only 3, or 3 with 1, or 3 with 2, or 3 with both 1 and 2. Since there are 2 elements remaining, each with 2 possibilities (in or out), there are \(2^2 = 4\) subsets containing the number 3: 1. {3} 2. {1, 3} 3. {2, 3} 4. {1, 2, 3}

Step 3: Count the subsets containing either the number 2 or the number 3

To count the subsets containing either the number 2 or the number 3, we can simply list all subsets from Step 1 and remove those that contain neither 2 nor 3. The remaining subsets are: 1. {2} 2. {3} 3. {1, 2} 4. {1, 3} 5. {2, 3} 6. {1, 2, 3} There are 6 subsets containing either the number 2 or the number 3.

Key Concepts

Sample Space

At its most fundamental, set theory begins with the concept of a sample space, which is the foundation of probability and counting. A sample space refers to the complete set of all possible outcomes that can occur in a particular experiment or situation. In set notation, a sample space is often denoted by the letter 'S', as in the given example where the sample space is defined as \(S=\{1,2,3\}\).

Understanding the sample space is crucial because it establishes the context within which events are considered, and it is the reference point for all probability calculations. If the sample space is clear and well-defined, students will find it easier to tackle problems related to probability and set theory.

Subset Counting

A subset is any selection of elements that forms a set, whether it includes all, some, or none of the original set's elements. The concept of subset counting involves determining the number of possible subsets for a given set. The number of subsets of a set with 'n' elements is found using the formula \(2^n\), because each element can either be included or excluded from a subset.

A strategy to make this concept easier for students to understand is to illustrate it visually or with a step-by-step process to showcase all the subsets, as shown in the provided solution. For instance, with 3 elements in set 'S', there are \(2^3 = 8\) possible subsets. This process shows that understanding powers of 2 is essential in subset counting.

Elementary Set Operations

Elementary set operations include the basic ways in which sets can be combined or related, such as union, intersection, and difference. In the context of set theory, an operation might ask a student to find elements common to specific sets (intersection) or to compile elements from multiple sets together (union). This particular exercise asks for subsets that contain the number 3, showcasing the use of selection criteria to form subsets, akin to an operation.

Incorporating exercises that involve selecting particular elements based on conditions can greatly improve students' comprehension of how set operations work. It encourages them to think about sets not just as static collections of numbers, but as entities that can be manipulated and explored under different conditions and rules.

Applied Mathematics

Set theory and its concepts are not abstract constructs; they are the building blocks for applied mathematics, including areas like probability, statistics, and data science. When students learn how to count subsets or perform elementary set operations, they're gaining skills that will help them analyze real-world problems where grouping and analyzing datasets is necessary. For example, understanding how to count the number of possible outcomes or events contributes directly to calculating probabilities.

Moreover, knowing set operations bolsters logical thinking and problem-solving abilities, which are invaluable in applied mathematics where one often needs to sift through data to find patterns or to make informed decisions based on given criteria. Exercises like those provided in the textbook help bridge the gap between theoretical math and practical application.