Indicate the least upper bound of the following sets If a se

Indicate the least upper bound of the following sets. If a set does not have a least upper bound, just write “none.” Needed for each set. I have no idea on how to find these

(2,3)

(2,3]

(1,]

Solution

Let S be a nonempty set of real numbers that has an upper bound. Then a number c is called the least upper bound (or the supremum, denoted sup S) for s iff it satisfies the following properties:

1. c>=x for all x belongs to s.

2. For all real numbers k, if k is an upper bound for S, then k >= c.

(2,3) - least upper bound is 3.

(2,3] - least upper bound is 3.

Q -None- the set Q of rational numbers does not have the least-upper-bound property under the usual order.

Q (1,] - None - does not have a least upper bound in Q.