True or False and why aIf S is a set of real numbers and M a

True or False, and why

(a)If S is a set of real numbers and M and m are upper and lower bounds of
S, respectively, then M and m are unique.

(b) If S is a bounded set of real numbers and M* and m* are least upper and
greatest lower bounds of S, respectively, then M* and m* are unique.

(d) If S is a bounded set of real numbers, then glb(S) < lub(S).

(e) If S is a bounded set of real numbers, then glb(S) < lub(S).

(f) Suppose that S is a bounded set of positive, real numbers, and set

Solution

a) True.

The upper bound and Lower bound need not belong to the set. upper bound is the Maximum element of set. then it is unique. similar argument can be made about lower bound.

b) True

If S is bounded set of real numbers by LUB axioms it has least upper bound and R has no holes LUB and GLB holds. hence it is unique

c) True

BY LUB property result is true

d)True

By LUB Property