1873_solutions

Given a subset g of r prove that the following two

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Unformatted text preview: e way in the form x C  y where C is an equivalence class of ß and y G. Solution: Suppose that x is any real number. We define C to be the equivalence class of the relation r such that x C. Since x C C we know that the ordered pair x, x C must belong to the relation r and so x x C G. Since x  xC  x xC we have shown that there exists an equivalence class C of r and a member y of the set G such that x  x C  y. Now we need to show that this decomposition of x into the form x C  y is unique. Suppose that C and D are both equivalence classes of r and that y and z are both members of the set G and that x C  y  x D  z. Since xC xD  z y G we see that the ordered pair x C , x D belongs to the relation r and we conclude that C  D. Therefore x  xC  y  xC  z and it follows that y  z. 11. Given that S is the set of all nonzero real numbers and that G 45 S and that x G, y what properties must the set G have in order to make r an equivalence relation in S? This exercise is similar to Exercise 9. The set G must be nonempty and whenever x and y belong to G, the numbers x 1 and xy must belong to G. r x, y S S 12. Suppose that P is the set of all positive numbers that are unequal to 1 and that Q is (as usual) the set of all rational numbers. a. Prove that the relation r defined by r x, y P P log x y Q is an equivalence relation in P. Given any number x P we have log x x  1 Q and so x r x. Therefore r is reflexive. Given numbers x and y in P it follows from the equation log x y  1 log y x that log x y Q if and only if log y x Q. Therefore r is symmetric. Given x, y and z in P, if the numbers log x y and log y z are rational then since log y z  log y z log x y Q log x z  log y x we conclude that r is transitive. b. Suppose that E is a subset of P that contains precisely one member of each equivalence class of the relation r. Prove that if x is any positive number unequal to 1 then x can be expressed in one and only one way in the form x  y q where y E and q is a rational number. We deduce from Exercise 6 that whenever x P there is exactly one member y of the set E such that log y x is rational. 13. Suppose that G is a set of nonzero real numbers and that P is the set of all positive numbers that are unequal to 1. Prove that the following two conditions are equivalent: a. The relation r defined by r x, y P P log x y G is an equivalence relation in P. b. The set G is nonempty and for all numbers a and b in the set G the number a b belongs to G. This exercise can be completed by the methods that were used in Exercises 9 and 11. An alternative is to use the conclusion of Exericise 11 as follows: We define W  log 2 x x P G. and we define the relation s in W by saying that if u and v belong to W then u s v means that u v Note that W is the set of nonzero real numbers. Since the condition u s v holds exactly when 2 u r 2 v we see at once that r is an equivalence relation in P if and only if s is an equivalence relation in the set W, the desired result follo...
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