1873_solutions

Belongs to then the union of any two members of

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Unformatted text preview: and observe that x A v is rational. In x is rational . c. Prove that if A and B are any two different members of then A B  . Suppose that A and B are members of and that A B. We may assume, without loss of generality that A B . Choose x A B. Given any member Y of the set B we know from part a of this exercise and from the fact that x B that the number x y can’t be rational; and therefore we know that y x can’t be rational. Thus no member y of the set B can belong to A and we have shown that A B  . d. Suppose that we have selected exactly one number in each member A of and have then collected these numbers together to form a set E. Suppose that for each rational number r we have defined Er  r  x x E . Prove that Þ Er r Q  R and that whenever r and s are different rational numbers then E r E s  . Suppose that u is any real number. Using the fact that Þ  R we choose a member A of such that u A. We know that the set E has exactly one member that belongs the set A and we call this member x. Since u  x u x and since u x is rational we know that u E u x . Finally suppose that r and s are rational numbers and that E r E s . We shall show that r  s. Choose a number u E r E s . Choose x and y in the set E such that u  r  x  s  y. Since x y  s r Q we know that x and y must belong to the same member of the family and therefore, from the way in which the set E was specified we conclude that x  y. The equation r  x  s  y now guarantees that r  s, as promised. Exercises on Relations 1. Under what conditions do we have x, y  y, x ? The condition x, y  y, x implies that x  y and is obviously true in this case. 2. For each of the relations defined earlier, find its domain and the range. a. We write P for the set of all people and define P r  x, y P x is a brother of y . The domain of r is the set of all those males who have at least one sibling. The range of r is the set of all those people who have at least one male sibling. b. We write P for the set of all people and define PP r  x, y x is a blood relation of y . c. We suppose that J is a bag (set) of jelly beans and define J J x and y have the same same color . r  x, y Both the domain and range of r are the entire set J. d. Writing R for the set of all real numbers, we define 42 r  x, y RR Both the domain and range of r are the entire set R. e. Writing Z for the set of all integers, we define ZZ r  x, y x y. Z such that y  nx . n Both the domain and range of r are the entire set Z. f. We define r x, y R R x y is rational . Both the domain and range of r are the entire set R. g. We define r  0, 1 , 0, 2 , 4, 1 , 3, 2 . Note that r is a subset of 0, 4, 3 1, 2 . The domain of r is 0, 4, 3 and the range of r is 1, 2 . h. We define r x, y R R x2  y2  1 . where R is the set of all real numbers. Given any numbers x and y, the equation x 2  y 2  1 requires that 1 Both the domain and range of r are equal to 1, 1 . x 1 and 1 y 1. i. Suppose that S is any set and define r  x, y...
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