Let R be the relation consisting of all pairs (x,y) such that x and y are strings of uppercase and lowercase English letters with the property that for every positive integer n, the nth characters in x and y are the same letter, either uppercase or lowercase. Show that R is an equivalence relation.

Answer:An equivalence relation R is a binary relation that is reflexive, symmetric and transitive.Step-by-step explanation:An equivalence relation R is a binary relation that is reflexive, symmetric and transitive.Reflexive:R is said to b reflexive if a R aSymmetric:R is said to be symmetric if a R b implies b R aTransitive:R is said to be transitive if a R b, b R c implies a R cGiven: Let R be the relation consisting of all pairs (x,y) such that x and y are strings of uppercase and lowercase English letters with the property that for every positive integer n, the nth characters in x and y are the same letter, either uppercase or lowercase.To prove: R is an equivalence relation.Reflexive:As the nth characters in x and x are the same letter, R is reflexiveSymmetric:If  nth characters in x and y are the same letter then clearly nth characters in y and x are the same letterTransitive:If  nth characters in x and y are the same letter and nth characters in y and z are the same letter then nth characters in x and z are the same letter.So, R is an equivalence relation.