If x ε S a non-empty set of real numbers, thena)inf x = sup xb)inf x = -sup (-x)c)inf x = sup (-x)d)inf x = -sup (x)Correct answer is option 'A'. Can you explain this answer?

Question

If x &epsilon; S a non-empty set of real numbers, thena)inf x = sup xb)inf x = -sup (-x)c)inf x = sup (-x)d)inf x = -sup (x)Correct answer is option 'A'. Can you explain this answer?

Answer

Explanation:

Definition of infimum and supremum:
- The infimum of a set S, denoted by inf S, is the greatest lower bound of S.
- The supremum of a set S, denoted by sup S, is the least upper bound of S.

Relationship between infimum and supremum:
- For any non-empty set S of real numbers, if x is an element of S, then inf S ≤ x ≤ sup S for all x in S.

Proof:
- Let x be an element of S.
- Since inf S is a lower bound of S, inf S ≤ x. This implies that sup S is an upper bound of the set (-x) = {-y | y ∈ S}.
- Therefore, -sup (-x) is the greatest lower bound of (-x).
- Since -sup (-x) = -inf x, we have -inf x ≤ -x for all x in S.
- Multiplying both sides by -1 gives inf x ≥ x for all x in S.
- Combining inf S ≤ x and inf x ≥ x, we have inf x = sup x.
Therefore, the correct answer is option 'A': inf x = sup x.

Answer

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