1873_solutions

Subsets of r describe the sets and a 11 n is a

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Unformatted text preview: that is a nested family of sets and that x Þ and y Þ , prove that there exists a member A of such that both x and y belong to A. Using the fact that x and y belong to Þ we choose a member A of such that x A and a member B of such that y B. Since is nested we know that either A B or B A and in either event we have found a member of to which both x and y belong. 4. A family of subsets of a set X is said to be field of subsets of X if to we have X A and A Þ B . and for all sets A and B that belong a. Prove that if is a field of subsets of a set X and A and B are sets that belong to , then the set A belongs to . Hint: Given any two subsets A and B of X we have 40 B BX A AÞX X B . b. Prove that if is a field of subsets of a set X and A and B are sets that belong to , then the set A belongs to . Given two subsets A and B of X we have A BA X B. B c. Prove that if is a collection of fields of subsets of a set X then the family is also a field of subsets of X. Since every member of contains we know that the family Þ also contains . Given any member A of we know that A belongs to every member of and therefore, since every member of is a field, X A belongs to every member of , and therefore belongs to . Finally, if A and B belong to then, since both A and B belong to every member of , so does A Þ B, which tells us that A Þ B . 5. Suppose that is a family of subsets of a set X and that X A is defined by A . a. Prove that if the intersection of any two members of belongs to then the union of any two members of belongs to . Suppose that the intersection of any two members of must belong to . Suppose that A and B belong to the family . Since both X A and X B belong to we know that XA XB and therefore X AÞB  X A XB from which we conclude that A Þ B . b. Prove that if  then Þ  X. We assume that  . To prove that Þ  X, suppose that x we choose a member A of such that x A. Since xXA we know that x Þ . c. Given that for every subfamily that . 1 Solution: Suppose that 1 1 we have Þ of 1 X. Using the fact that x , and given that 1 is a subfamily of , prove is a subfamily of . We define X A A 1 , we have Þ 1 . Therefore Þ1 . 1X 1 and we observe that, since 1 6. This exercise will explore some of the properties of the family that was defined in an earlier example. Recall that is defined to be the family of all those subsets A of R for which it is possible to find a real number a such that A  x R x a is a rational number . a. Suppose that A is a member of the family and only if u v is rational. and that v A. Prove that a real number u will belong to A if Solution: Choose a number a such that A Since v x R x a is a rational number . A we know that the number v a is rational. Now given any number u we see that u a u v  v a 41 and we deduce that the number u a will be rational if and only if the number u other words, the condition u A holds if and only if u v is rational. b. Prove that Þ  R. Suppose that x is any real number. We define A t R t...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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