Problem 38 - Discrete Mathematics with Applications

Question

Find the family of subsets of each set that do not contain consecutive integers. $$\\{1,2\\}$$

Step-by-Step Solution

Answer

The family of subsets of the set \([1, 2]\) that do not contain consecutive integers are: \(\emptyset\), \(\{1\}\), and \(\{2\}\).

Step 1: Find all subsets of the set [1, 2]

We can find all subsets of the set \([1, 2]\) using the power set method. A set with \(n\) elements has \(2^n\) subsets. In this case, the given set has 2 elements, so it has \(2^2 = 4\) subsets. The subsets are: 1. \(\emptyset\) (the empty set) 2. \(\{1\}\) 3. \(\{2\}\) 4. \(\{1, 2\}\)

Step 2: Filter out the subsets with consecutive integers

Now, we want to filter out the subsets that contain consecutive integers. Since the original set only has two elements, the only subset that can potentially contain consecutive integers is \(\{1, 2\}\). The other three subsets do not have the possibility of consecutive integers because they either contain no elements or just one element. Therefore, after filtering out the subset with consecutive integers, we are left with the following family of subsets: 1. \(\emptyset\) 2. \(\{1\}\) 3. \(\{2\}\)

Key Concepts

Subsets

Understanding subsets is essential for solving many mathematical problems. A subset is a selection of elements from a set, where each element is either included or excluded. This results in multiple possible combinations.
For example, consider the set \([1, 2]\). It contains two elements. Using the concept of power sets, we find all possible subsets. The power set of a set with \(n\) elements has \(2^n\) subsets. Here, since \(n = 2\), we have:

• The empty set: \(\emptyset\)
• A single-element subset: \(\{1\}\)
• Another single-element subset: \(\{2\}\)
• The entire set itself: \(\{1, 2\}\)

If you're not familiar with subsets or power sets, think of it like choosing outfits from a set of clothes—you can take some, all, or none!

Consecutive Integers

Consecutive integers are numbers that follow each other without gaps. They are like stepping stones on a path. In the set \([1, 2]\), the numbers 1 and 2 are consecutive.
Recognizing consecutive integers in subsets is crucial because they behave differently in certain mathematical contexts. For instance:

• If you pick \(\{1, 2\}\), you've picked consecutive numbers.
• On the other hand, \(\{1\}\) or \(\{2\}\) individually does not contain any consecutive integers.

Consecutive integers have special properties and considerations, which often means filtering them out for specific problems.

Filtering Subsets

Filtering subsets involves removing specific elements based on given criteria. In this problem, the criterion is the presence of consecutive integers.
Here's what we did:

• Start with the power set: \(\emptyset, \{1\}, \{2\}, \{1, 2\}\).
• Identify the subset with consecutive integers: \(\{1, 2\}\).
• Exclude this subset to meet the problem's requirements.
• The resulting subsets are: \(\emptyset, \{1\}, \{2\}\).

Filtering helps to simplify problems and focus on specific characteristics needed for further analysis. It is like skimming the cream from milk—keeping only the parts you need.