Problem 3 - Applied Discrete Structures

Question

List all two-element sets in \(\mathcal{P}(\\{a, b, c, d\\})\)

Step-by-Step Solution

Answer

The two-element subsets of \( \{a, b, c, d\} \) are \( \{a, b\} \), \( \{a, c\} \), \( \{a, d\} \), \( \{b, c\} \), \( \{b, d\} \), and \( \{c, d\} \).

Step 1: Understand the Power Set

The power set of a set is the set of all its subsets. Given a set \( \{a, b, c, d\} \), its power set, denoted \( \mathcal{P}(\{a, b, c, d\}) \), includes every possible subset, including the empty set and the set itself.

Step 2: Identify Two-Element Subsets

To form all two-element subsets from the set \( \{a, b, c, d\} \), we need to choose two elements at a time. We can express this as combinations: \( \binom{n}{r} = \binom{4}{2} \), where \( n = 4 \) is the total number of elements and \( r = 2 \) is the subset size.

Step 3: Calculate Combinations

Calculate the number of two-element combinations possible using the formula for combinations: \( \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \cdot 3}{2 \cdot 1} = 6 \). This tells us there are 6 unique two-element subsets.

Step 4: List All Two-Element Subsets

Enumerate all two-element subsets by selecting pairs:- \( \{a, b\} \)- \( \{a, c\} \)- \( \{a, d\} \)- \( \{b, c\} \)- \( \{b, d\} \)- \( \{c, d\} \)These are all the subsets that contain exactly two elements.

Key Concepts

Power Set

A power set is an important concept in combinatorics that refers to the set of all possible subsets of a given set. When we talk about subsets, we mean groups that can be formed using elements from our original set. For example, let's consider the set \( \{a, b, c, d\} \). Its power set, denoted as \( \mathcal{P}(\{a, b, c, d\}) \), includes every possible subset.
- This ranges from the smallest subset, which is the empty set \( \{\} \), to the largest subset, which is the set itself \( \{a, b, c, d\} \).
- There will be smaller sets, too, like one-element subsets \( \{a\}, \{b\}, \{c\}, \{d\} \), and so forth.
Understanding the power set helps us grasp the wealth of combinations that can be formed, and it's foundational for tackling problems involving subsets and combinations.

Subsets

Subsets are central when studying power sets and combinations. A subset contains zero or more elements from the original set. It's simply a way of grouping these elements.
For the set \( \{a, b, c, d\} \), some subsets include:

• The empty set \( \{\} \)
• Single-element subgroups like \( \{a\} \) or \( \{b\} \)
• Two-element combinations such as \( \{a, b\} \) or \( \{c, d\} \)

Every subset is counted in the power set, which means both small and large groupings are possible. The power set guarantees we analyze every configuration available within the elements of the original set.

Combinations

Combinations are a mathematical method used for counting or listing the ways in which a set of items can be arranged without the order being significant. The concept is particularly useful in finding subsets of a certain size.
To find how many two-element subsets exist within the set \( \{a, b, c, d\} \), we use combinations expressed as \( \binom{n}{r} \), which reads as 'n choose r'.
- Here, \( n \) is 4 (total elements in our set), and \( r \) is 2 (the size of the subset we're interested in).
- This calculates to \( \binom{4}{2} \), leading to \( \frac{4!}{2!(4-2)!} = 6 \), meaning six possible two-element subsets.
Using combinations helps simplify the process of listing subsets, as seen in the enumeration of \( \{a, b\}, \{a, c\}, \{a, d\}, \{b, c\}, \{b, d\}, \{c, d\} \). It shows you not only how many subsets there are, but also identifies them clearly, making it an invaluable tool in combinatorial mathematics.