Instructions:Show that the given relation R is an equivalence relation on set S.

Question

Instructions:Show that the given relation R is an equivalence relation on set S. Then describe the equivalence class containing the the given element z in S, and determine the number of distinct equivalence classes of R.-------Let S be the set of all possible strings of 3 or 4 letters, let z=ABCD, and define x R y to mean that x has the same first letter as y and also the same third letter as y. ----------- I'm honestly baffled as to how to approach this problem, let alone the solution. Please explain how to set up/solution and the reasoning behind it. Thank you!!

Explanation / Answer

Equivalence relation means the relation works like equality in some ways: x~x always as x = ABCD will always have the same first and third letters as x x~y => y~x as if x has same first and third letter as y then y will have same first and third letter as x x~y and y~z => x~z as if x has same first and third letter as y and if y has same first and third letter as z then x will have same first and third letter as z Hence the relation ~ on all 3-4 letter strings is reflexive, symmetric and transitive. ie it is an equivalence relation. the equivalence class of a string z=ABCD = A$C# where $ and # can be any letter. It is the set of strings with the same starting and third letter as z

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