Getting Close to Sets

We will use tex2html_wrap_inline303 to denote the set of real numbers and tex2html_wrap_inline305 to denote the set of natural numbers. Sometimes real numbers are called scalars. Whenever A is a set, we will write tex2html_wrap_inline309 to say ``x is an element of A.'' If A and B are sets we will write tex2html_wrap_inline319 to say ``A is a subset of B.'' If A and B are sets then we define

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We say that a set A is finite if there is a natural number n and elements tex2html_wrap_inline333 in A for every natural number tex2html_wrap_inline337 so that

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For example, the set tex2html_wrap_inline339 is a finite set, but tex2html_wrap_inline305 is not a finite set.

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  1. Suppose tex2html_wrap_inline431 and tex2html_wrap_inline433
    1. Show that tex2html_wrap_inline435
    2. Is there a smallest positive number tex2html_wrap_inline437 so that tex2html_wrap_inline439 Prove your claim.
  2. Suppose tex2html_wrap_inline441 and tex2html_wrap_inline443 Show that tex 2html_wrap_inline445 and tex2html_wrap_inline447
  3. Show that if A is a nonempty finite subset of tex2html_wrap_inline303 and B is any subset of R then tex2html_wrap_inline445
  4. Give examples of subsets A and B of tex2html_wrap_inline303 to show that tex2html_wrap_inline445 does not necessarily imply tex2html_wrap_inline447
  5. Let t ex2html_wrap_inline469 and suppose tex2html_wrap_inline471 such that tex2html_wrap_inline473 and tex2html_wrap_inline475 Let tex2html_wrap_inline477 denote the union of A and C, and let tex2html_wrap_inline483 denote the intersection of A and C. That is

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    and

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    1. Show tex2html_wrap_inline489
    2. Show tex2html_wrap_inline491 provided tex2html_wrap_inline493 is not the empty set.
  6. Give examples of subsets A, B and C of tex2html_wrap_inline303 to show that tex2html_wrap_inline445 and tex2html_wrap_inline505 does not necessarily imply tex2html_wrap_inline507
  7. Give examples of subsets A, B and C of tex2html_wrap_inline303 to show th at tex2html_wrap_inline445 and tex2html_wrap_inline519 does not necessarily imply tex2html_wrap_inline507
  8. Let tex2html_wrap_inline523 denote the set of rational numbers and tex2html_wrap_inline525 denote the set of irrational numbers. It is a well know property of the real numbers that a rational number lies between any two distinct real numbers. Show that tex2html_wrap_inline527 Then use the property above to prove tex2html_wrap_inline529 and tex2html_wrap_inline531
  9. If tex2html_wrap_inline533 we define

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    Show that if tex2html_wrap_inline535 and tex2html_wrap_inline445 then tex2html_wrap_inli ne539

  10. Let tex2html_wrap_inline535 and suppose tex2html_wrap_inline543 Define

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    and

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    1. Show that if A and B satisfy tex2html_wrap_inline445 then tex2html_wrap_inline551 .
    2. Show by examp le that if tex2html_wrap_inline553 satisfy tex2html_wrap_inline555 and tex2html_wrap_in line557 then it is not necessarily the case that

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  11. Let tex2html_wrap_inline559 and define

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    and

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    (Note: The argument tex2html_wrap_inline561 above is given in radians, not degrees!!)

    1. Show that if tex2html_wrap_inline559 then tex2html_wrap_inline565 .
    2. Suppose tex2html_wrap_inline559 is a rational number. Is there a smallest positive number tex2html_wrap_inline437 so that tex2html_wrap_inline571 Prove your claim.
    3. Show that if tex2html_wrap_inline559 is an irrational number then tex2html_wrap_inline575
    4. Show that if tex2html_wrap_inline577 are irrational numbers then tex2html_wrap_inline579
  12. Let tex2html_wrap_inline581 and tex2html_wrap_inline583 Is there a smallest possible number tex2html_wrap_inline471 so that

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    Prove your claim.

  13. Define

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    Let P and J be points in tex2html_wrap_inline591 . We define

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    Then, inductively, for n=2,3,... define

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    Similarly, we define

    B1={J}

    Then, inductively, for n=2,3,... define

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    Let tex2html_wrap_inline597 and tex2html_wrap_inline599 Show that tex2html_wrap_inline 601 Notes: tex2html_wrap_inline603 is the midpoint of the line segment joining U and V. The choice tex2html_wrap_inline609 leads to the set A shown below.

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  14. Define tex2html_wrap_inline613 , tex2html_wrap_inline615 and tex2html_wrap_inline617 For each tex2html_wrap_inline619 let tex2html_wrap_inline621 be chosen from the set tex2html_wrap _inline623 and form the number a given by the decimal expansion

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    Now create a set A as follows. Let tex2html_wrap_inline629 be a point in tex2html_wrap_inline631 Set

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    and inductively, for n=2,3,4,... define

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    Then set tex2html_wrap_inline635

    1. Show that if a is a rational number then there exist a finite number of points tex2html_wrap_inline639 in tex2html_wrap_inline591 so that

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    2. Is the converse of the statement above true?
    3. Let B be a set constructed in the manner described in problem 13. Is it possible for the values tex2html_wrap_inline621 to be chosen in such a way that tex2html_wrap_inline647