MAJORANT - MINORANT - SUPREMUM - INFIMUM - MAXIMUM - MINIMUM

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K which is greater than or equal to every element of S

Dually, a lower bound or minorant of S is defined to be an element of K which is less than or equal to every element of S

A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (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

A ⊆ ℝ, A ≠ Ø, s ∈ ℝ, s = upper bound or majorant, a ≤ s ∀ a ∈ A

A ⊆ ℝ, A ≠ Ø, s ∈ ℝ, s = lower bound or minorant, a ≥ s ∀ a ∈ A

A = [1,3) = {x ∈ ℝ | 1 ≤ x < 3}, 3 = upper bound or majorant, 3 ≥ a ∀ a ∈ A

A = [-1,4] = {x ∈ ℝ | -1 ≤ x ≤ 4}, -2 = lower bound or minorant, -2 ≤ a ∀ a ∈ A

A = [-1,4] = {x ∈ ℝ | -1 ≤ x ≤ 4}, -1 = lower bound or minorant, -1 ≤ a ∀ a ∈ A

A = ℕ, ∀ s ∈ ℝ, ∃ n ∈ ℕ | s < n ⇒ ℕ does not have a majorant or upper bound

The minorant or lower bound of ℕ is 0

Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention)

An infinite subset of the natural numbers cannot be bounded from above

ℤ is not bounded above or below

ℤ \ ℕ is limited above but not below; 0 is a majorant or upper bound

(-∞,1] is limited above but not below

Definition of majorant or upper bound: A ⊆ ℝ, s ∈ ℝ, s ≥ a ∀ a ∈ A

Definition of minorant or lower bound: A ⊆ ℝ, s ∈ ℝ, s ≤ a ∀ a ∈ A

A = [0,1), s ≥ 1 = majorant or upper bound of A, s < 1 ≠ majorant or upper bound of A

A = (-1,5], s ≤ -1 = minorant or lower bound of A, s > -1 ≠ minorant or lower bound of A

A ⊆ ℝ, A ≠ Ø, S ∈ ℝ is the supremum of A if it is the least upper bound of A

S = sup A ⇔ {S ≥ a ∀ a ∈ A; ∀ ε > 0, ∃ a ∈ A | S - ε < a}

S = sup A ⇔ {S ≥ a ∀ a ∈ A (S = majorant or upper bound of A); ∀ ε > 0, ∃ a ∈ A | S - ε < a (S - ε ≠ majorant or upper bound of A)}

A ⊆ ℝ, A ≠ Ø, s ∈ ℝ is the infimum of A if it is the greatest lower bound of A

s = inf A ⇔ {s ≤ a ∀ a ∈ A; ∀ ε > 0, ∃ a ∈ A | s + ε > a}

s = inf A ⇔ {s ≤ a ∀ a ∈ A (s = minorant or lower bound of A); ∀ ε > 0, ∃ a ∈ A | s + ε > a (s + ε ≠ minorant or upper bound of A)}

Supremum is the least upper bound; infimum is the greatest lower bound

Definition of supremum: A ⊆ ℝ, A ≠ Ø, s = sup A ⇔ {s ≥ a ∀ a ∈ A; ∀ ε > 0 ∃ a ∈ A | s - ε < a}

Definition of infimum: A ⊆ ℝ, A ≠ Ø, i = inf A ⇔ {i ≤ a ∀ a ∈ A; ∀ ε > 0 ∃ a ∈ A | i + ε > a}

A = [0,1) = {x ∈ ℝ | 0 ≤ x < 1}; sup A = 1 (1 = majorant or upper bound); ∀ ε > 0 ∃ x ∈ [0,1) | 1 - ε < x (1 = least upper bound = supremum)

A = [0,1) = {x ∈ ℝ | 0 ≤ x < 1}; inf A = 0 (0 = minorant or lower bound); ∀ ε > 0 ∃ x ∈ [0,1) | 0 + ε > x (0 = greatest lower bound = infimum)

If the supremum belongs to the set, then it is the maximum of the set

A ⊆ ℝ, A ≠ Ø, if sup A ∈ A, then sup A = max A

If the infimum belongs to the set, then it is the minimum of the set

A ⊆ ℝ, A ≠ Ø, if inf A ∈ A, then inf A = min A

Intuitively, completeness implies that there are not any "gaps" (in Dedekind's terminology) or "missing points" in the real number line; this contrasts with the rational numbers, whose corresponding number line has a “gap” at each irrational value

In mathematics, the least-upper-bound property (sometimes called completeness or supremum property) is a fundamental property of the real numbers

An ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X

Not every ordered set has the least upper bound property; for example, the set ℚ of all rational numbers with its natural order does not have the least upper bound property

Completeness of ℝ Theorem: ℝ is complete because every upper-bounded subset of ℝ admits a supremum, and every lower-bounded subset of ℝ admits an infimum

Completeness of ℝ Theorem: ℝ is complete because every upper-bounded subset of ℝ admits a supremum (that is the least upper bound), and every lower-bounded subset of ℝ admits an infimum (that is the greatest lower bound)

The subset A is upper-bounded if it admits an upper bound or majorant

The subset A is lower-bounded if it admits a lower bound or minorant

The property of completeness is true in ℝ, but is false in ℚ

A = {x ∈ ℝ | x^{2} < 2} = (-√2,√2); sup A = √2, inf A = -√2; √2 ∈ ℝ \ ℚ

A = {x ∈ ℚ | x^{2} < 2} = (-√2,√2) ∩ ℚ; sup A ≠ √2 (√2 ∉ ℚ); inf A ≠ -√2 (-√2 ∉ ℚ)

Density of ℚ in ℝ: ∀ x,y ∈ ℝ, x < y, ∃ q ∈ ℚ | x < q < y

Considering the property of the rational numbers, the density of ℚ in ℝ, it is impossible to find a supremum or an infimum, because there is always a rational number between two real numbers; there is no completeness in ℚ

The property of completeness is false in ℚ; in the set of rational numbers the sumpremum (the least upper bound or least majorant) does not exist, and the infimum (the greatest lower bound or greatest minorant) does not exist

(ℚ, +, ⋅, ≥); ℚ is not complete because every subset of ℚ does not admit a supremum or an infimum

(ℝ, +, ⋅, ≥); ℝ is complete because every subset of ℝ admits a supremum or an infimum

If A ≠ Ø and it is upper-bounded, S = Sup A always exists (completeness); if S ∈ A, then S = sup A = max A

A = (1,2), sup A = 2 and 2 ∉ A then ∄ max A, inf A = 1 and 1 ∉ A then ∄ min A

If A is not upper-bounded, sup A = +∞

If A is not lower-bounded, inf A = -∞

Sup ℕ = +∞

Inf ℤ = -∞

Sup ℤ = +∞

A = [-π,3] ∪ {0} ∪ [1,+∞], inf A = -π = min A, sup A = +∞

A = {1/n | n ∈ ℕ \ {0}}; A = {1\n | n ∈ ℕ}; sup A = max A = 1; inf A = 0; 0 ≤ 1/n ∀ n ∈ ℕ \ {0}; ∀ ε > 0 ∃ n ∈ ℕ | 0 + ε > 1/n; ε > 1/n ⇔ n > 1/ε; if 0 = min A ⇔ 0 ∈ A ⇔ ∃ n ∈ ℕ | 0 = 1/n and it does not admit solutions; 0 ≠ 1/n

A = {1 - 1/n | n ∈ ℕ \ {0}}; {0, 1/2, 2/3, 3/4, 4/5...}; 0 ≤ 1 - 1/n ≤ 1; inf A = 0, 0 ∈ A, min A = 0; sup A = 1, 1 - 1/n ≤ 1 ∀ n ∈ ℕ (1 is an upper bound, a majorant), ∀ ε > 0 ∃ n ∈ ℕ \ {0} | 1 - ε < 1 - 1/n (1 is the least upper bound, the supremum, ∀ ε > 0 ∃ a ∈ A | S - ε < a), 1 - ε < 1 - 1/n ⇔ - ε < - 1/n ⇔ ε > 1/n ⇔ n > 1/ε, sup A = 1; if max A = 1 ⇔ ∃ n ∈ ℕ \ {0} | 1 - 1/n = 1 ⇔ - 1/n = 0 (never)

A = {n + 2/n | n ∈ ℕ}; 0 ≤ n + 2/n (0 is a minorant or lower bound); n + 2/n ≥ n ∀ n ∈ ℕ, sup A = +∞ (no majorant or upper bound); n = 1: 1 + 2/1 = 3; n = 2: 2 + 2/2 = 3; n = 3: 3 + 2/3 > 3; n = 4: 3 + 2/4 > 3; ∀ n ≥ 3, n + 2/n ≥ n ≥ 3; min A = 3; A = {n + 2/n | n = 1, n = 2} ∪ {n + 2/n | n ≥ 3} = {3} ∪ {n + 2/n | ∀ n ≥ 3}