Question

List all possible arrangements of the four letters \(m, a, r\), and \(y .\) Let \(C_{1}\) be the collection of the arrangements in which \(y\) is in the last position. Let \(C_{2}\) be the collection of the arrangements in which \(m\) is in the first position. Find the union and the intersection of \(C_{1}\) and \(C_{2}\).

Short answer

The possible arrangements of the four letters \(m, a, r, y\) are 24. The collections \(C_{1}\) and \(C_{2}\) depend on the specific arrangements. The union of \(C_{1}\) and \(C_{2}\) includes all unique arrangements in the two collections. The intersection of \(C_{1}\) and \(C_{2}\) includes only those arrangements that appear in both collections.

Step-by-step solution

Arranging the letters

Firstly, all possible arrangements of the four letters \(m, a, r, y\) are needed to be listed. Since there are 4 different letters, there are \(4!\) or 24 possible arrangements.

Finding the specific collections

Now, two specific collections need to be found. Collection \(C_{1}\) contains arrangements where \(y\) is in the last position. Collection \(C_{2}\) includes arrangements where \(m\) is in the first position. These collections can be found by looking at the arrangements and selecting those that meet the criteria.

Union of Collections

The union of \(C_{1}\) and \(C_{2}\) is the set of all arrangements that are in \(C_{1}\), in \(C_{2}\), or in both. It can be found by listing all unique arrangements in the two collections.

Intersection of Collections

The intersection of \(C_{1}\) and \(C_{2}\) is the set of all arrangements that are both in \(C_{1}\) and in \(C_{2}\). It can be found by listing only those arrangements that appear in both collections.

Key concepts

Permutations of Letters

When we talk about permutations of letters, we're diving into the fascinating world of combinatorial arrangements. Permutation is a mathematical term that refers to the number of ways a set of objects can be rearranged without any repeats. In the case of letters, each different sequence constitutes a unique permutation.

For instance, if we have the letters 'm', 'a', 'r', and 'y', we approach the problem by considering each letter as an individual item that can occupy any position in a sequence. The total number of permutations of four distinct letters is calculated using factorial notation, which for four items is represented as '4!'. The formula for the number of permutations of 'n' distinct objects is expressed as 'n! = n × (n - 1) × (n - 2) ... × 1'. Hence, '4!' equals '4 × 3 × 2 × 1', resulting in 24 unique permutations.

It's key to remember that permutations are sensitive to the order, so 'mary' and 'army' would be considered different arrangements. Understanding this concept lays the groundwork for analyzing more complex combinatorial problems.

Set Theory in Combinatorics

Set theory serves as the backbone of combinatorics. It provides a way to describe collections of objects, called sets, and allows us to perform operations like union and intersection on these collections. When we group certain permutations of letters together based on specific conditions, we are effectively creating sets of those permutations.

In our exercise, we define two sets based on where certain letters appear. Collection C₁ is a set containing all permutations where the letter 'y' is in the last position. Similarly, collection C₂ is the set of all permutations with 'm' in the first position. These conditions filter the total permutations into smaller, well-defined groups. By applying set theory, we derive a structured approach to analyzing these groups and identifying their relationships.

• Union (C₁ ∪ C₂): A combination of all unique elements from both sets.
• Intersection (C₁ ∩ C₂): Only the elements that appear in both sets.

Set theory terminology is extremely useful for expressing these relationships clearly and succinctly. Understanding these concepts allows us to handle and solve complex combinatorial challenges.

Union and Intersection of Sets

The union and intersection of sets are fundamental operations in set theory that reveal distinct relationships between sets. When presented with two sets, finding their union means determining all distinct elements present in at least one of the sets. On the other hand, finding their intersection requires checking for elements that both sets have in common.

Specifically, for the exercise involving permutations of the letters 'm', 'a', 'r', and 'y', the union of sets C₁ and C₂ results in a set including all permutations where 'y' is last or 'm' is first, without duplication. The intersection consists solely of permutations that satisfy both conditions simultaneously.

Translating this to a practical approach, one might list all the permutations in C₁, then all in C₂, and then combine these lists, ensuring not to include any permutation more than once for the unionC₁ ∪ C₂. For the intersection C₁ ∩ C₂, the focus would be on finding permutations common to both initial sets. This clear method of categorization allows for an organized way to solve problems related to the combination of different sets.