# Equivalence Classes and such

Let A be the set of all possible strings of 3 or 4 letters in alphabet ${A,B,C,D}$, let z = $BCAD$, and let $(x,y)\in R$ if and only if $x$ and $y$ have the same first letter and the same third letter

I have already confirmed that this is an equivalence relation and now I have to find the equivalence classes, the number of equivalence classes and determine which equivalence class belongs to Z when Z= $BCAD$

I have no idea where to start. (Textbook question, not for homework)

Please do not tell me the answer, but tell me how to begin or give a portion of the answer.

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To give a concrete example of an equivalence class, suppose we have the string $ACBD$. What are all the members of it's equivalence class? Well to be in the same equivalence class as $ACBD$, we need the string to have the same first letter as $ACBD$, namely $A$, and the same third letter, namely $B$. Then the strings are of the form $A?B?$. Now we have to fill in the question marks. Each choice of letters will be fine. $$AABA,\ AABB,\ AABC,\ AABD,\ AAB$$ $$ABBA,\ ABBB,\ ABBC,\ ABBD,\ ABB$$ $$ACBA,\ ACBB,\ ACBC,\ ACBD,\ ACB$$ $$ADBA,\ ADBB,\ ADBC,\ ADBD,\ ADB$$ And those are all the members in the equivalence class of $ACBD$. You would do something similar to find out the members of the equivalence class for $BCAD$.