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- Definitions: A set is bounded above by the number A if the number A is higher than or equal to all elements of the set. A set is bounded below by the number B if the number B is lower than or equal to all elements of the set.
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A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S . The concepts of bounded below and lower bound are defined similarly.
Oct 2, 2020 · A clear explanation of sets bounded from above and from below, upper bounds, and lower bounds. Insightful examples that show how to prove that a set is bounded.
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- Big Epsilon
A is bounded above (or right bounded) iff there is q ∈ F such that. (∀x ∈ A) x ≤ q. In this case, p and q are called, respectively, a lower (or left) bound and an upper (or right) bound, of A. If both exist, we simply say that A is bounded (by p and q).
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
1. Your proof is completely correct: nice! In a general ordered set S S (like that of the integers (Z, ≤) (Z, ≤)), we have that " A ⊆ S A ⊆ S is bounded above" takes the following meaning: There is an s ∈ S s ∈ S such that for all a ∈ A: a ≤ s a ∈ A: a ≤ s.
A set A ⊂ Rof real numbers is bounded from above if there exists a real number M ∈ R, called an upper bound of A, such that x ≤ M for every x ∈ A. Similarly, A is bounded from below if there exists m ∈ R, called a lower bound of A, such that x ≥ m for every x ∈ A. A set is bounded if it is bounded both from above and below. The ...
Theorem 1: If $A \subseteq B$ and $B$ is a bounded set then $A$ is a bounded set. Further, if $A \subseteq B$ and $A$ is an unbounded set then $B$ is an unbounded set.
