next up previous
Next: Sequence Limits Up: Elementary Analysis Previous: Elementary Analysis

Boundedness

  1. In the following, let S be a subset of the real numbers and B be a real number. Rewrite each statement symbolically using quantifiers.
    1. Definition: B is an upper bound for S if tex2html_wrap_inline159 for all tex2html_wrap_inline161 .
    2. Definition: B is the least upper bound for S if B is an upper bound for S and if B' is any upper bound for S, then tex2html_wrap_inline175 .
    3. Definition: B is a lower bound for S if tex2html_wrap_inline181 for all tex2html_wrap_inline161 .
    4. Definition: B is the greatest lower bound for S if B is a lower bound for S and if B' is any lower bound for S, then tex2html_wrap_inline197 .
    5. Definition: B is a bound for S if tex2html_wrap_inline203 for all tex2html_wrap_inline161 .
    6. Definition: S is a bounded set if it has a bound.
    7. Least Upper Bound Axiom for the Real Numbers: If S is nonempty and has an upper bound, then it has a least upper bound.

    1. Express the following statements symbolically.
      1. B is not an upper bound for S.
      2. S has no upper bound.
      3. B is not a least upper bound for S.
    2. Prove or disprove.
      1. The empty set has an upper bound.
      2. The empty set has a least upper bound.
      3. If S is a nonempty subset of the real numbers and S has a lower bound, then it has a greatest lower bound.
      4. If S is a nonempty subset of the rational numbers that has an upper bound, then it has a rational least upper bound.
      5. The number 3 is an upper bound for the set tex2html_wrap_inline229 .
      6. The number 3 is an upper bound for the set tex2html_wrap_inline233 .
      7. The set tex2html_wrap_inline235 has an upper bound.
      8. The set tex2html_wrap_inline233 has an upper bound.


Carl Lee
Wed Dec 2 12:11:38 EST 1998