Home Homework Library Mathematics Advanced Math 1. Suppose that A âŠ† B and that B has a greatest lower bound. ...

1. Suppose that A âŠ† B and that B has a greatest lower bound. Show...

Question

1. Suppose that A âŠ† B and that B has a greatest lower bound. Show that A has a greatest lower bound and that glbB â‰¤ glbA.

2. Let A, B âŠ† â„ (â„ represents the real numbers) and A, B both non-empty. Define A + B = {a+b: aâˆˆA and bâˆˆB}. Suppose A and B both have least upper bounds. Prove that the set A + B then also has a least upper bound and that lub(A + B) = lubA + lubB.

3. Designate each of the following sets as ordered fields or not. Type F if the given set is an ordered field and type N if it is not.
a. â„ (the real numbers)
b. â„š (the rational numbers)
c. â„‚ (the complex numbers)
d. â„• (the natural numbers)
e. â„ - â„š

4. Suppose set A has a greatest lower bound. Then it is not necessarily true that lub(-A) = -glbA.
True or False

5. Set A is bounded below if and only if set â€“A is bounded above.
True or False

6. Set A is bounded above if and only if set â€“A is bounded below.
True or False

7. Let A âŠ† â„ (â„ represents the real numbers) and A non-empty. Prove TWO of the following:
a. glbA â‰¤ lubA
b. Let x,y âˆˆ A. Then |x â€“ y|â‰¤ lubA â€“ glbA
c. glbA = lubA if and only if A contains only a single element

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1. Let aâˆˆA (we assume A is non-empty). Then since AâŠ†B, then aâˆˆB and thus aâ‰¥glb B. By the Greatest Lower Property, A has a greatest lower bound. Since glb B is a lower bound of A, then as desired we must have glb Aâ‰¥glb B by definition.

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