Problem 3 - A Gentle Introduction to the Art of Mathematics

Question

Define a relation A on the set of all words by \(w_{1} \mathrm{~A} w_{2} \Longleftrightarrow w_{1}\) is an anagram of \(w_{2}\) Show that A is an equivalence relation. (Words are anagrams if the letters of one can be re-arranged to form the other. For example, 'ART" and 'RAT' are anagrams.)

Step-by-Step Solution

Answer

Relation A is an equivalence relation because it is reflexive, symmetric, and transitive.

Step 1 - Prove Reflexivity

To prove reflexivity, we need to show that any word is an anagram of itself. Let’s take any word 'w'. Since the word 'w' contains the same letters as itself when rearranged, we get that \( w \text{ A } w \).Thus, relation A is reflexive.

Step 2 - Prove Symmetry

To prove symmetry, we need to show that if one word is an anagram of another, then the latter is also an anagram of the first. Let’s take two words 'w1' and 'w2' such that \( w_{1} \text{ A } w_{2} \). This means that 'w1' is an anagram of 'w2'.Therefore, 'w2' must also contain the same letters as 'w1' rearranged.Thus, \( w_{2} \text{ A } w_{1} \) and relation A is symmetric.

Step 3 - Prove Transitivity

To prove transitivity, we need to show that if one word is an anagram of a second, and the second is an anagram of a third, then the first and third words are anagrams.Let’s take three words 'w1', 'w2', and 'w3' such that \( w_{1} \text{ A } w_{2} \) and \( w_{2} \text{ A } w_{3} \). This means 'w1' is an anagram of 'w2', and 'w2' is an anagram of 'w3'.Thus, 'w1' and 'w3' must have the same letters rearranged.Therefore,\( w_{1} \text{ A } w_{3} \) and relation A is transitive.

Key Concepts

Reflexivity

Reflexivity is a property of a relation that requires every element to be related to itself. In the context of anagrams, this means that any given word is an anagram of itself. Let's break it down: Take any word 'w'. Clearly, 'w' always contains the same letters as itself, regardless of how those letters are rearranged. Thus, we can confidently state that \( w \text{ A } w \). This confirms that relation A is reflexive. Here, 'A' means the word is an anagram of itself.

Symmetry

Symmetry in a relation means that if one element is related to a second element, then the second element must also be related to the first. For anagrams, if one word is an anagram of another, then the reverse must also be true.
To illustrate this, consider two words 'w1' and 'w2'. Suppose \( w_{1} \text{ A } w_{2} \), which means 'w1' is an anagram of 'w2'. Therefore, 'w2' also has the same letters rearranged as 'w1'.
Consequently, we can assert that \( w_{2} \text{ A } w_{1} \). This proves that relation A is symmetric.

Transitivity

Transitivity is a property demanding that if one element is related to a second, and the second is related to a third, then the first element must also be related to the third. For anagrams, this means if one word is an anagram of a second word, and the second word is an anagram of a third word, then the first and third words must also be anagrams.
Let's take three words 'w1', 'w2', and 'w3'. Assume \( w_{1} \text{ A } w_{2} \) and \( w_{2} \text{ A } w_{3} \). This implies that 'w1' is an anagram of 'w2', and 'w2' is an anagram of 'w3'. Hence, 'w1' and 'w3' contain the same letters arranged differently.
Therefore, we can conclude that \( w_{1} \text{ A } w_{3} \), confirming that relation A is transitive.