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## 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 for all .
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 .
3. Definition: B is a lower bound for S if for all .
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 .
5. Definition: B is a bound for S if for all .
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 .
6. The number 3 is an upper bound for the set .
7. The set has an upper bound.
8. The set has an upper bound.

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