Let A={a, b, …, z} be the set consisting of the letters of...

Let $A=\{a, b, \ldots, z\}$ be the set consisting of the letters of the alphabet. For $\alpha \in A$, let $A_\alpha$ consist of $\alpha$ and the two letters that follow it, where $A_y=\{y, z, a\}$ and $A_z=\{z, a, b\}$. Find a set $S \subseteq A$ of smallest cardinality such that $\bigcup_{\alpha \in S} A_\alpha=A$. Explain why your set $S$ has the required properties.

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Key Concepts

Set Covering Problem

This concept involves selecting a collection of subsets from a given set of subsets such that their union equals the entire universal set. In such problems, we seek the smallest number of these subsets that together cover all elements, which is key to optimizing and solving problems efficiently in combinatorics and computer science.

Cyclic Structures

A cyclic structure is one where the elements are arranged in a circle, so that the end of the sequence wraps around to the beginning. This concept is essential for understanding problems that include wrap?around or rotation, as it affects how subsets overlapping near the boundaries are formed and counted.

Modular Arithmetic

Modular arithmetic deals with numbers wrapping around after reaching a certain value, known as the modulus. In the context of cyclic structures, it underlies the method of computing succeeding elements, especially when the sequence reaches its endpoint and starts over, ensuring a continuous cycle.

Cardinality

Cardinality refers to the number of elements in a set. In optimization problems, such as finding a minimal cover, reducing the cardinality of the chosen subset is crucial. It measures the efficiency of the solution, indicating the smallest number of subsets needed to cover the entire set.

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