Define a relation on R{0} (that is, the set of real numbers with 0 removed) as...

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Define a relation on R\{0} (that is, the set of real numbers with 0 removed) as follows: ~ "~y" means that EQ. (Recall that is the set of rational numbers.) Prove (using the format demonstrated in class) that is an equivalence relation on R\{0}. Write up a nice version of your argument instead of submitting scratch work. (CAUTION: The prescription that "Q" is NOT meant to imply that x and y must be integers-or even that they must be rational! For instance, we have √2~3√2-check this using the definition!-even though √2 and 3√2 are both irrational.) Prove the following: For any positive integer a and any integers x and n, if x =, n, then ged(a, n) |x. Define a relation on R\{0} (that is, the set of real numbers with 0 removed) as follows: ~ "~y" means that EQ. (Recall that is the set of rational numbers.) Prove (using the format demonstrated in class) that is an equivalence relation on R\{0}. Write up a nice version of your argument instead of submitting scratch work. (CAUTION: The prescription that "Q" is NOT meant to imply that x and y must be integers-or even that they must be rational! For instance, we have √2~3√2-check this using the definition!-even though √2 and 3√2 are both irrational.) Prove the following: For any positive integer a and any integers x and n, if x =, n, then ged(a, n) |x.

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