Question

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 1

The answer is RR and Rr..... Only if this was your question which is down below under explanation

Explanation:

Analyze the table below and answer the question that follows.

A square with four boxes. Uppercase R lowercase r are on the left side of the square. Uppercase R lowercase r are on the top side of the square. In the square: Uppercase R uppercase R in the first box on the top. Uppercase R lowercase r in the second box on the top. Uppercase R lowercase r in the first box on the bottom. Lowercase r lowercase r in the second box on the bottom.

If the ability to roll one’s tongue is dominant (R) and the inability to roll one’s tongue is recessive (r), what are the possible genotypes for someone who can roll their tongue?

only RR

RR and Rr

RR and rr

only rr

Answer 2

An equivalence relation R is a binary relation that is reflexive, symmetric and transitive.

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.