Exercise · ZW-SUPREMUM-AND-INFIMUM-008 · ★★☆ Intermediate
If \(S=(1,4)\) and \(-S=\{-s : s\in S\}\), what is \(\sup(-S)\)?
Choose one option.
Show answer
- A. \(-1\)
Explanation
We have \(\sup(-S)=-\inf S=-1\); indeed \(-S=(-4,-1)\).
Review the concept: Supremum and infimum
Given a set \(S\) of real numbers, its supremum is the least of its upper bounds and its infimum is the greatest of its lower bounds.
\[s\le\sup S,\quad s\ge\inf S\qquad\forall s\in S\]
More practice problems: supremum and infimum
- ZW-SUPREMUM-AND-INFIMUM-001 What is the supremum of the interval (0,1)?
- ZW-SUPREMUM-AND-INFIMUM-002 What is the infimum of [2,7)?
- ZW-SUPREMUM-AND-INFIMUM-003 What is the maximum of (0,1)?
- ZW-SUPREMUM-AND-INFIMUM-004 Find sup\1-1/n : n∈mathbb N\ with n≥1.
- ZW-SUPREMUM-AND-INFIMUM-005 Find inf\1/n : n∈mathbb N\ with n≥1.
- ZW-SUPREMUM-AND-INFIMUM-006 Find sup\x∈mathbb R : x^2<2\.
- ZW-SUPREMUM-AND-INFIMUM-007 What is the supremum of (0,1)∪[2,3)?
- ZW-SUPREMUM-AND-INFIMUM-009 Let M be an upper bound of S. When is M=sup S?