Question

Let \(E\) be the set of letters in the word MATROIDS. Show that the family (STAR, ROAD, MOAT, RIOT, RIDS, DAMS, MIST) of subsets of \(E\) has exactly eight transversals.

Short answer

The family of subsets of \(E\), (STAR, ROAD, MOAT, RIOT, RIDS, DAMS, MIST), has exactly eight transversals. The enumeration and counting require careful and systematic exploration

Step-by-step solution

Understanding Transversals

In set theory, a transversal (or system of distinct representatives) for a family of sets is a set containing exactly one element from each of the given sets. In our case, each set represents a subset of letters in the word 'MATROIDS'. So, a transversal will be a selection of letters so that each letter is from a different word/subset and all subsets have at least one letter represented.

Enumerate Transversals

Given that the assignments of a letter to a word can also be seen as a permutation of those words, the problem of finding transversals here should be solved by considering the various ways letters can be assigned to the words, such that each word and each letter is used exactly once. This can be done as follows: Start by assigning M to MOAT, then the only option left for A is STAR, S can be assigned to MIST or DAMS, T can be assigned to MIST or RIOT. R can be assigned to RIOT, STAR or ROAD, D can be assigned to ROAD or DAMS, I can be assigned to RIOT and O can be assigned to ROAD or MOAT. By systematically assigning letters to words and ensuring no letter or word is repeated, the transversals can be enumerated by branching at the choices and backtracking if a dead end (i.e. any letter or word has no available assignment) is reached.

Count Transversals

Keeping track of these assignments and making sure that each word receives a different letter each time results in exactly 8 different transversals that can be made. These are the unique sets of assignments as a result of permutations.

Key concepts

Understanding Transversals

In the realm of set theory, a transversal is a fascinating concept. It's a set that picks exactly one element from each of a given family of sets. Imagine you have a collection of different baskets, and you need to pick one fruit from each basket. That collection of fruits represents a transversal. In our example, MATROIDS forms the "fruit baskets" (subsets of letters), and "fruits" are individual letters. A transversal ensures each subset is represented just once. This teaches us how distinct choices from collections interplay in forming complete representations.

System of Distinct Representatives

A system of distinct representatives (SDR) focuses on ensuring there's no overlap in selections. Think of it as choosing class representatives where each class must be represented by a unique student. Our challenge involves assigning letters uniquely to subsets such as STAR, ROAD, and so on. Each subset should get exactly one letter that isn't repeated in another one. It's about precision in selection, aiming for fairness and distinctiveness across the represented sets.

How to Enumerate Transversals

Enumerating transversals is about counting available combinations. Imagine arranging books on a shelf, where each must fit perfectly without overlap. We start with one letter, say 'M' with MOAT, and follow the permissible paths, ensuring each word receives a distinct one. Systematic choice follows, where each letter placement leaves further options, exploring these paths one by one. It’s like a puzzle where each piece carefully slots into the correct gap, making sure each possibility is explored without repetition.

Permutations in the Process

Permutation is the science of arranging elements methodically. In our problem, assigning different letters to words involves permutations, where order and allocation become crucial. Permutations help kickstart identifying feasible transversals, as swapping letters around alters outcomes. Each feasible arrangement (or permutation) devises a unique transversal. This reflects the variety in ways letters fit into subsets, leading us to discover all the possible transversals when permutations are concluded effectively.

Set of Subsets and Their Structure

Dealing with a set of subsets like STAR, ROAD, MOAT, etc., is like working with nested layers. Each subset represents a unique perspective on the overarching set 'MATROIDS'. Here, structure and hierarchy play a key role, as transversals are formed by picking elements across layers. The subsets provide the framework, while transversals are the detailed filling. By understanding sets dynamically, one can innovate new ways to assemble elements efficiently, adhering to constraints like uniqueness and coverage—all crucial for solutions.