# The relation R on R (all real #s) given by x R y iff x - y element of Q?

prove that the relation is an equivalence relation.

Then give information about the equivalence classes as specified.

The relation R on R (all real #s) given by x R y iff x - y element of Q. Give the Equivalence class of 0; of 1, of √2.

Relevance

Let R be a relation on R(all reals) given by x R y iff x - y is in Q. To show R is an equivalence relation, then we must show that R is reflexive, symmetric, and transitive.

Reflexive: We must show for all x in R, that x R x. Let x be in R. Then x - x = 0 which is clearly in Q. Thus, x R x and so R is reflexive.

Symmetric: We must show for all x, y in R that if x R y, then y R x. Let x and y be in R and assume that x R y. Thus, x - y is in Q. Note that Q is closed under addition and multiplication. (-1)(x - y) = y - x is in Q since -1 and x - y are in Q. Thus, y R x. Thus, R is symmetric.

Transitive: We must show for all x, y, z in R that if x R y and y R z, then x R z. Let x, y, z be in R and assume that x R y and y R z. Then x - y and y - z are in Q. Since Q is closed under addition, (x - y) + (y - z) = x - z is in Q. Thus, x R z. Thus, R is transitive.

Thus, R is an equivalence relation.

Let x be in R. Equivalence class of x = {y | x R y} = {y | x - y is in Q}

If x is in Q, then eqv class of x = Q.

If x is in R\Q, then eqv class of x = {a - x where a is in Q}