Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List all subsets of the following sets: a. \(\\{1,2\\}\) b. \(\\{1,2,3\\}\) c. \(\\{1,2,3,4\\}\)

Step-by-Step Solution

Answer

The subsets are: a. \(\\{\\}\), \(\\{1\\}\), \(\\{2\\}\), \(\\{1,2\\}\) b. \(\\{\\}\), \(\\{1\\}\), \(\\{2\\}\), \(\\{3\\}\), \(\\{1,2\\}\), \(\\{1,3\\}\), \(\\{2,3\\}\), \(\\{1,2,3\\}\) c. \(\\{\\}\), \(\\{1\\}\), \(\\{2\\}\), \(\\{3\\}\), \(\\{4\\}\), \(\\{1,2\\}\), \(\\{1,3\\}\), \(\\{1,4\\}\), \(\\{2,3\\}\), \(\\{2,4\\}\), \(\\{3,4\\}\), \(\\{1,2,3\\}\), \(\\{1,2,4\\}\), \(\\{1,3,4\\}\), \(\\{2,3,4\\}\), \(\\{1,2,3,4\\}\)

Step 1: Identify size of subsets

Start by listing subsets of increasing size (empty set, one-element subsets, and so on) until reaching the size of the entire set.

Step 2: List the subsets

List all subsets of the set \(\\{1,2\\}\): - Empty set: \(\\{\\}\) - One-element subsets: \(\\{1\\}\), \(\\{2\\}\) - Two-element subset: \(\\{1,2\\}\) b. List all subsets of the set \(\\{1,2,3\\}\)

Step 1: Identify size of subsets

In this case, we again start by listing subsets of increasing size (empty set, one-element subsets, etc.) until reaching the size of the entire set.

Step 2: List the subsets

List all subsets of the set \(\\{1,2,3\\}\): - Empty set: \(\\{\\}\) - One-element subsets: \(\\{1\\}\), \(\\{2\\}\), \(\\{3\\}\) - Two-element subsets: \(\\{1,2\\}\), \(\\{1,3\\}\), \(\\{2,3\\}\) - Three-element subset: \(\\{1,2,3\\}\) c. List all subsets of the set \(\\{1,2,3,4\\}\)

Step 1: Identify size of subsets

As before, we will start by listing subsets of increasing size (empty set, one-element subsets, etc.) until we reach the size of the entire set.

Step 2: List the subsets

List all subsets of the set \(\\{1,2,3,4\\}\): - Empty set: \(\\{\\}\) - One-element subsets: \(\\{1\\}\), \(\\{2\\}\), \(\\{3\\}\), \(\\{4\\}\) - Two-element subsets: \(\\{1,2\\}\), \(\\{1,3\\}\), \(\\{1,4\\}\), \(\\{2,3\\}\), \(\\{2,4\\}\), \(\\{3,4\\}\) - Three-element subsets: \(\\{1,2,3\\}\), \(\\{1,2,4\\}\), \(\\{1,3,4\\}\), \(\\{2,3,4\\}\) - Four-element subset: \(\\{1,2,3,4\\}\)

Key Concepts

Set Theory

Set theory is a fundamental concept in mathematics that deals with the study of collections of objects. These collections are called sets, and they can contain anything: numbers, letters, or even other sets. The power of set theory comes from its ability to formalize the notion of a 'collection' of objects and the relationships between these collections.

A key aspect of set theory is the concept of subsets. A subset is simply a set that contains only elements from another set, called the original or parent set. For example, if we have a parent set \(\{1, 2, 3\}\), subsets can include \(\{1\}\), \(\{2, 3\}\), and even the empty set, \(\{\}\), which contains no elements.

The process of finding all subsets involves making sure we cover every possible combination of elements from the set. This ties into the notion of the power set, which is the set of all subsets of a given set. The size of the power set of a set with \(n\) elements is \(2^n\). Each element in the set has two choices: to be included in a subset or not.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and grouping elements within a set. It provides the foundation for listing subsets in set theory, which is essentially about determining how many different ways elements of a set can be grouped.

When working with subsets, combinatorics kicks in to help us calculate the number of potential subsets. As mentioned earlier, a set with \(n\) elements will have \(2^n\) subsets. This is because each element has two possibilities: it can either be included in a subset or not. For example, a set \(\{1, 2\}\) has \(2^2 = 4\) subsets: \(\{\}\), \(\{1\}\), \(\{2\}\), and \(\{1, 2\}\).

This combination calculation is derived using the concept of binary counting, where every element has a "binary choice" of being included in a subset. This is a simplified form of the broader study of combinations, permutations, and other structural arrangements in combinatorics.

Mathematical Reasoning

Mathematical reasoning is the process of using logical steps and principles to solve problems and prove statements. In the context of finding subsets, mathematical reasoning allows us to systematically determine every possible subset of a given set.

The reasoning process involves:

• Recognizing the recursive pattern of creating subsets by adding one element at a time.
• Understanding that the empty set is always a subset of any set, as it contains no elements.
• Using binary thinking to decide whether each element in the original set is part of a subset or not.

For example, when listing subsets of \(\{1, 2, 3\}\), we start by considering the smallest subsets (the empty set and single-element subsets), then combine elements to form larger subsets.

The logic flows from smaller to larger subsets, ensuring all possibilities are captured without repetition. This structured approach is fundamental in mathematical reasoning, where problems are broken down into manageable parts and solved step-by-step.