Detailed solutions please! 2. (10 pts) For each of the following statements, det

Question

Detailed solutions please!

2. (10 pts) For each of the following statements, determine whether it is true or false. Give a counterexample IF IT IS FALSE (but you do not need to prove it if it is true). (a) A bounded above set of real numbers always has a maximum. (b) A nonempty bounded above set of real numbers always has a supremum. (c) A nonempty bounded above set of integers always has a supremum and a maximum. (d) If neither the set A nor the set B is dense in R, then AuB is not dense in R. (e) A set that is dense in R must have an infinite number of members. (f Aset that is dense in R does NOT have a supremum. (g) Let S (0.001 n n EZ). Then S is dense in R.

Explanation / Answer

A True
B True
C True
D False as if there is some intersection area between A and B then it might be denser
E False as it doesn't signify anything
F False it may have supremum

g False

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