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20 August, 11:07

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.

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  1. 20 August, 11:32
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    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.
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