Lower and Upper Bounds

2 Key excerpts on "Lower and Upper Bounds"

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  Mathematical Foundations of Fuzzy Sets

  • Hsien-Chung Wu(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  2 1 Mathematical Analysis Definition 1.1.2 Let S be a subset of ℝ. (i) Suppose that S is bounded above. A real number ̄ u ∈ ℝ is called a least upper bound or supremum of S when the following conditions are satisfied. ● ̄ u is an upper bound of S. ● If u is any upper bound of S, then u ≥ ̄ u. In this case, we write ̄ u = sup S. We say that the supremum sup S is attained when ̄ u ∈ S. (ii) Suppose that S is bounded below. A real number ̄ l ∈ ℝ is called a greatest lower bound or infimum of S when the following conditions are satisfied. ● ̄ l is a lower bound of S. ● If l is any lower bound of S, then l ≤ ̄ l. In this case, we write ̄ l = inf S. We say that the infimum inf S is attained when ̄ l ∈ S. It is clear to see that if the supremum sup S is attained, then max S = sup S. Similarly, if the infimum inf S is attained, then min S = inf S. Example 1.1.3 Let S = [0,1]. Then, we have max S = sup S = 1 and inf S = min S = 0.

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  Handbook of Mathematics for Engineers and Scientists

  • Andrei D. Polyanin, Alexander V. Manzhirov(Authors)
  • 2006(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  1. Sets of the form ( a , b ), (– ∞ , b ), ( a , + ∞ ), and (– ∞ , + ∞ ) consisting, respectively, of all x R such that a < x < b , x < b , x > a , and x is arbitrary are called open intervals (sometimes simply intervals ). 2. Sets of the form [ a , b ] consisting of all x R such that a ≤ x ≤ b are called closed intervals or segments . 3. Sets of the form ( a , b ], [ a , b ), (– ∞ , b ], [ a , + ∞ ) consisting of all x such that a < x ≤ b , a ≤ x < b , x ≤ b , x ≥ a are called half-open intervals . A neighborhood of a point x ◦ R is de fi ned as any open interval ( a , b ) containing x ◦ ( a < x ◦ < b ). A neighborhood of the “point” + ∞ , – ∞ , or ∞ is de fi ned, respectively, as any set of the form ( b , + ∞ ), (– ∞ , c ) or (– ∞ , – a ) ∪ ( a , + ∞ ) (here, a ≥ 0 ). 6.1.1-2. Lower and upper bound of a set on a straight line. The upper bound of a set of real numbers is the least number that bounds the set from above. The lower bound of a set of real numbers is the largest number that bounds the set from below. In more details: let a set of real numbers X R be given. A number β is called its upper bound and denoted sup X if for any x X the inequality x ≤ β holds and for any β 1 < β there exists an x 1 X such that x 1 > β 1 . A number α is called the lower bound of X and denoted inf X if for any x X the inequality x ≥ α holds and for any α 1 > α there exists an x 1 X such that x 1 < α 1 . Example 1. For a set X consisting of two numbers a and b ( a < b ), we have inf X = a , sup X = b . Example 2. For intervals (open, closed, and half-open), we have inf( a , b ) = inf[ a , b ] = inf( a , b ] = inf[ a , b ) = a , sup( a , b ) = sup[ a , b ] = sup( a , b ] = sup[ a , b ) = b . 235 236 L IMITS AND D ERIVATIVES One can see that the upper and lower bounds may belong to a given set (e.g., for closed intervals) and may not (e.g., for open intervals). The symbol + ∞ (resp., – ∞ ) is called the upper (resp., lower) bound of a set unbounded from above (resp., from below).

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