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.

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 a

Symmetric:

R is said to be symmetric if a R b implies b R a

Transitive:

R is said to be transitive if a R b, b R c implies a R c

Given: 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 reflexive

Symmetric:

If nth characters in x and y are the same letter then clearly nth characters in y and x are the same letter

Transitive:

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.