Answer to Question #244839 in Real Analysis for Hiruni

Question

Questions
1. Infimum of the set(0,00)
(a) is a non-negative number.
(b) is a positive number.
(c) does not exist.
(d) none of these.
2.Which of the following is not true for a set in R?
(a) A set may not have an infimum in R.
(b) Infimum of a set may not belong to the set.
(c) Infimum and supremum of a set may be equal.
(d) Supremum of a bounded below set always exists in R.
3.Which algebraic property is not true for the set of real numbers R?
(a) For all a

Answer 1

Solution:

(1):

Set (0,00).

Both elements are same, they denote 0.

Infimum = greatest lower bound.

Thus, infimum of (0,00) = 0, which is non-negative number.

So, option (a) is correct.

(2):

Let (a,b] be a set. Here infimum of this set is 'a'.

But 'a' does not belong to that set.

Thus, option (b) Infimum of a set may not belong to the set, is correct.