Search results
Results from the WOW.Com Content Network
If the supremum of exists, it is unique, and if b is an upper bound of , then the supremum of is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB). [1] The infimum is, in a precise sense, dual to the concept of a
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bound s, that is, = {: ({: <}) =} if the set of essential lower bounds is nonempty, and as otherwise; again there is an alternative expression as = {: ()} (with this being if the set is empty).
The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum. Interval. For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty. Interval ...
This concept is also called supremum or join, and for a set S one writes sup(S) or for its least upper bound. Conversely, the greatest lower bound is known as infimum or meet and denoted inf(S) or . These concepts play an important role in many applications of order theory.
The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.
Discover the best free online games at AOL.com - Play board, card, casino, puzzle and many more online games while chatting with others in real-time.
That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity ...