Misplaced Pages

#278721

84-619: The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis , and especially in Lebesgue integration . However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maximum , but are more useful in analysis because they better characterize special sets which may have no minimum or maximum . For instance,

168-396: A {\displaystyle a} of S {\displaystyle S} is called an infimum (or greatest lower bound , or meet ) of S {\displaystyle S} if Similarly, an upper bound of a subset S {\displaystyle S} of a partially ordered set ( P , ≤ ) {\displaystyle (P,\leq )}

252-525: A R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. In mathematics , specifically order theory , the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle S,} denoted ⋁ S , {\textstyle \bigvee S,} and similarly,

336-408: A lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties . If the supremum of

420-420: A (possibly different) non-increasing sequence s 1 ≥ s 2 ≥ ⋯ {\displaystyle s_{1}\geq s_{2}\geq \cdots } in S {\displaystyle S} such that lim n → ∞ s n = inf S . {\displaystyle \lim _{n\to \infty }s_{n}=\inf S.} Expressing

504-428: A least element, then that element is the infimum; otherwise, the infimum does not belong to S {\displaystyle S} (or does not exist). The infimum of a subset S {\displaystyle S} of a partially ordered set P , {\displaystyle P,} assuming it exists, does not necessarily belong to S . {\displaystyle S.} If it does, it

588-431: A least upper bound, then S {\displaystyle S} is said to have the least-upper-bound property. As noted above, the set R {\displaystyle \mathbb {R} } of all real numbers has the least-upper-bound property. Similarly, the set Z {\displaystyle \mathbb {Z} } of integers has the least-upper-bound property; if S {\displaystyle S}

672-438: A meet, then the meet can still be seen as a partial binary operation on A . {\displaystyle A.} If the meet does exist then it is denoted x ∧ y . {\displaystyle x\wedge y.} If all pairs of elements from A {\displaystyle A} have a meet, then the meet is a binary operation on A , {\displaystyle A,} and it

756-453: A mnemonic for remembering that ∨ {\displaystyle \,\vee \,} denotes the join/supremum and ∧ {\displaystyle \,\wedge \,} denotes the meet/infimum ). More generally, suppose that F ≠ ∅ {\displaystyle {\mathcal {F}}\neq \varnothing } is a family of subsets of some set X {\displaystyle X} that

840-457: A partially ordered set in which all pairs have a meet is a meet-semilattice . A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice . A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice . It is also possible to define a partial lattice , in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms. The join/meet of

924-508: A point p , {\displaystyle p,} then f ( s 1 ) , f ( s 2 ) , … {\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots } necessarily converges to f ( p ) . {\displaystyle f(p).} It implies that if lim n → ∞ s n = sup S {\displaystyle \lim _{n\to \infty }s_{n}=\sup S}

SECTION 10

#1732855348279

1008-509: A set For any set S {\displaystyle S} that does not contain 0 , {\displaystyle 0,} let 1 S   := { 1 s : s ∈ S } . {\displaystyle {\frac {1}{S}}~:=\;\left\{{\tfrac {1}{s}}:s\in S\right\}.} If S ⊆ ( 0 , ∞ ) {\displaystyle S\subseteq (0,\infty )}

1092-1206: A set The product of a real number r {\displaystyle r} and a set B {\displaystyle B} of real numbers is the set r B   :=   { r ⋅ b : b ∈ B } . {\displaystyle rB~:=~\{r\cdot b:b\in B\}.} If r ≥ 0 {\displaystyle r\geq 0} then inf ( r ⋅ A ) = r ( inf A )  and  sup ( r ⋅ A ) = r ( sup A ) , {\displaystyle \inf(r\cdot A)=r(\inf A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\sup A),} while if r ≤ 0 {\displaystyle r\leq 0} then inf ( r ⋅ A ) = r ( sup A )  and  sup ( r ⋅ A ) = r ( inf A ) . {\displaystyle \inf(r\cdot A)=r(\sup A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\inf A).} Using r = − 1 {\displaystyle r=-1} and

1176-407: A subset S {\displaystyle S} exists, it is unique. If S {\displaystyle S} contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S {\displaystyle S} (or does not exist). Likewise, if the infimum exists, it is unique. If S {\displaystyle S} contains

1260-686: A subset S {\displaystyle S} in P {\displaystyle P} equals the supremum of S {\displaystyle S} in P op {\displaystyle P^{\operatorname {op} }} and vice versa. For subsets of the real numbers, another kind of duality holds: inf S = − sup ( − S ) , {\displaystyle \inf S=-\sup(-S),} where − S := { − s   :   s ∈ S } . {\displaystyle -S:=\{-s~:~s\in S\}.} In

1344-493: A subset S {\displaystyle S} of ( N , ∣ ) {\displaystyle (\mathbb {N} ,\mid \,)} where ∣ {\displaystyle \,\mid \,} denotes " divides ", is the lowest common multiple of the elements of S . {\displaystyle S.} The supremum of a set S {\displaystyle S} containing subsets of some set X {\displaystyle X}

1428-743: A subset S {\displaystyle S} of P {\displaystyle P} can fail if S {\displaystyle S} has no lower bound at all, or if the set of lower bounds does not contain a greatest element. (An example of this is the subset { x ∈ Q : x 2 < 2 } {\displaystyle \{x\in \mathbb {Q} :x^{2}<2\}} of Q {\displaystyle \mathbb {Q} } . It has upper bounds, such as 1.5, but no supremum in Q {\displaystyle \mathbb {Q} } .) Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance,

1512-441: A subset need not be members of that subset themselves. Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least"

1596-415: A subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists. If a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is also an (upward) directed set , then its join (if it exists) is called a directed join or directed supremum . Dually, if S {\displaystyle S}

1680-761: A supremum, this applies also, for any set X , {\displaystyle X,} in the function space containing all functions from X {\displaystyle X} to P , {\displaystyle P,} where f ≤ g {\displaystyle f\leq g} if and only if f ( x ) ≤ g ( x ) {\displaystyle f(x)\leq g(x)} for all x ∈ X . {\displaystyle x\in X.} For example, it applies for real functions, and, since these can be considered special cases of functions, for real n {\displaystyle n} -tuples and sequences of real numbers. The least-upper-bound property

1764-1087: Is empty , one writes inf S = + ∞ . {\displaystyle \inf _{}S=+\infty .} If A {\displaystyle A} is any set of real numbers then A ≠ ∅ {\displaystyle A\neq \varnothing } if and only if sup A ≥ inf A , {\displaystyle \sup A\geq \inf A,} and otherwise − ∞ = sup ∅ < inf ∅ = ∞ . {\displaystyle -\infty =\sup \varnothing <\inf \varnothing =\infty .} If A ⊆ B {\displaystyle A\subseteq B} are sets of real numbers then inf A ≥ inf B {\displaystyle \inf A\geq \inf B} (unless A = ∅ ≠ B {\displaystyle A=\varnothing \neq B} ) and sup A ≤ sup B . {\displaystyle \sup A\leq \sup B.} Identifying infima and suprema If

SECTION 20

#1732855348279

1848-1369: Is partially ordered by ⊆ . {\displaystyle \,\subseteq .\,} If F {\displaystyle {\mathcal {F}}} is closed under arbitrary unions and arbitrary intersections and if A , B , ( F i ) i ∈ I {\displaystyle A,B,\left(F_{i}\right)_{i\in I}} belong to F {\displaystyle {\mathcal {F}}} then A ∨ B = A ∪ B , A ∧ B = A ∩ B , ⋁ i ∈ I F i = ⋃ i ∈ I F i ,  and  ⋀ i ∈ I F i = ⋂ i ∈ I F i . {\displaystyle A\vee B=A\cup B,\quad A\wedge B=A\cap B,\quad \bigvee _{i\in I}F_{i}=\bigcup _{i\in I}F_{i},\quad {\text{ and }}\quad \bigwedge _{i\in I}F_{i}=\bigcap _{i\in I}F_{i}.} But if F {\displaystyle {\mathcal {F}}}

1932-532: Is a meet if it satisfies the three conditions a , b , and c . The pair ( A , ∧ ) {\displaystyle (A,\wedge )} is then a meet-semilattice . Moreover, we then may define a binary relation ≤ {\displaystyle \,\leq \,} on A , by stating that x ≤ y {\displaystyle x\leq y} if and only if x ∧ y = x . {\displaystyle x\wedge y=x.} In fact, this relation

2016-455: Is a minimum or least element of S . {\displaystyle S.} Similarly, if the supremum of S {\displaystyle S} belongs to S , {\displaystyle S,} it is a maximum or greatest element of S . {\displaystyle S.} For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of

2100-399: Is a partial order on A . {\displaystyle A.} Indeed, for any elements x , y , z ∈ A , {\displaystyle x,y,z\in A,} Both meets and joins equally satisfy this definition: a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as

2184-469: Is a corresponding greatest-lower-bound property ; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set. If in a partially ordered set P {\displaystyle P} every bounded subset has

2268-450: Is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum . Let A {\displaystyle A} be a set with a partial order ≤ , {\displaystyle \,\leq ,\,} and let x , y ∈ A . {\displaystyle x,y\in A.} An element m {\displaystyle m} of A {\displaystyle A}

2352-451: Is a meet defined by the partial order defined by the original meet, and the two meets coincide. In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relation and a binary operation, such that each one of these structures determines the other, and fulfill the conditions for partial orders or meets, respectively. If ( A , ∧ ) {\displaystyle (A,\wedge )}

2436-403: Is a meet-semilattice, then the meet may be extended to a well-defined meet of any non-empty finite set, by the technique described in iterated binary operations . Alternatively, if the meet defines or is defined by a partial order, some subsets of A {\displaystyle A} indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of

2520-503: Is a nonempty subset of Z {\displaystyle \mathbb {Z} } and there is some number n {\displaystyle n} such that every element s {\displaystyle s} of S {\displaystyle S} is less than or equal to n , {\displaystyle n,} then there is a least upper bound u {\displaystyle u} for S , {\displaystyle S,} an integer that

2604-865: Is a real number (where all s 1 , s 2 , … {\displaystyle s_{1},s_{2},\ldots } are in S {\displaystyle S} ) and if f {\displaystyle f} is a continuous function whose domain contains S {\displaystyle S} and sup S , {\displaystyle \sup S,} then f ( sup S ) = f ( lim n → ∞ s n ) = lim n → ∞ f ( s n ) , {\displaystyle f(\sup S)=f\left(\lim _{n\to \infty }s_{n}\right)=\lim _{n\to \infty }f\left(s_{n}\right),} which (for instance) guarantees that f ( sup S ) {\displaystyle f(\sup S)}

Infimum and supremum - Misplaced Pages Continue

2688-577: Is an a ϵ ∈ A {\displaystyle a_{\epsilon }\in A} with a ϵ < p + ϵ . {\displaystyle a_{\epsilon }<p+\epsilon .} Similarly, if sup A {\displaystyle \sup A} is a real number and if p {\displaystyle p} is any real number then p = sup A {\displaystyle p=\sup A} if and only if p {\displaystyle p}

2772-443: Is an adherent point of the set f ( S ) = def { f ( s ) : s ∈ S } . {\displaystyle f(S)\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\{f(s):s\in S\}.} If in addition to what has been assumed, the continuous function f {\displaystyle f} is also an increasing or non-decreasing function , then it

2856-423: Is an element z {\displaystyle z} of P {\displaystyle P} such that An upper bound b {\displaystyle b} of S {\displaystyle S} is called a supremum (or least upper bound , or join ) of S {\displaystyle S} if Infima and suprema do not necessarily exist. Existence of an infimum of

2940-407: Is an indicator of the suprema. In analysis , infima and suprema of subsets S {\displaystyle S} of the real numbers are particularly important. For instance, the negative real numbers do not have a greatest element, and their supremum is 0 {\displaystyle 0} (which is not a negative real number). The completeness of the real numbers implies (and

3024-494: Is an upper bound and if for every ϵ > 0 {\displaystyle \epsilon >0} there is an a ϵ ∈ A {\displaystyle a_{\epsilon }\in A} with a ϵ > p − ϵ . {\displaystyle a_{\epsilon }>p-\epsilon .} Relation to limits of sequences If S ≠ ∅ {\displaystyle S\neq \varnothing }

3108-449: Is an upper bound for S {\displaystyle S} and is less than or equal to every other upper bound for S . {\displaystyle S.} A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set. An example of a set that lacks the least-upper-bound property is Q , {\displaystyle \mathbb {Q} ,}

3192-551: Is another lower bound of x  and  y , {\displaystyle x{\text{ and }}y,} then w ∧ x = w ∧ y = w , {\displaystyle w\wedge x=w\wedge y=w,} whence w ∧ z = w ∧ ( x ∧ y ) = ( w ∧ x ) ∧ y = w ∧ y = w . {\displaystyle w\wedge z=w\wedge (x\wedge y)=(w\wedge x)\wedge y=w\wedge y=w.} Thus, there

3276-475: Is any non-empty set of real numbers then there always exists a non-decreasing sequence s 1 ≤ s 2 ≤ ⋯ {\displaystyle s_{1}\leq s_{2}\leq \cdots } in S {\displaystyle S} such that lim n → ∞ s n = sup S . {\displaystyle \lim _{n\to \infty }s_{n}=\sup S.} Similarly, there will exist

3360-403: Is called the meet (or greatest lower bound or infimum ) of x  and  y {\displaystyle x{\text{ and }}y} and is denoted by x ∧ y , {\displaystyle x\wedge y,} if the following two conditions are satisfied: The meet need not exist, either since the pair has no lower bound at all, or since none of

3444-570: Is captured by the Duality Principle for ordered sets: If a statement or definition is equivalent to its dual then it is said to be self-dual . Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol. Naturally, there are a great number of examples for concepts that are dual: Examples of notions which are self-dual include: Since partial orders are antisymmetric ,

Infimum and supremum - Misplaced Pages Continue

3528-825: Is defined similarly to their Minkowski sum: A ⋅ B   :=   { a ⋅ b : a ∈ A , b ∈ B } . {\displaystyle A\cdot B~:=~\{a\cdot b:a\in A,b\in B\}.} If A {\displaystyle A} and B {\displaystyle B} are nonempty sets of positive real numbers then inf ( A ⋅ B ) = ( inf A ) ⋅ ( inf B ) {\displaystyle \inf(A\cdot B)=(\inf A)\cdot (\inf B)} and similarly for suprema sup ( A ⋅ B ) = ( sup A ) ⋅ ( sup B ) . {\displaystyle \sup(A\cdot B)=(\sup A)\cdot (\sup B).} Scalar product of

3612-532: Is easy to see that this operation fulfills the following three conditions: For any elements x , y , z ∈ A , {\displaystyle x,y,z\in A,} Joins are defined dually with the join of x  and  y , {\displaystyle x{\text{ and }}y,} if it exists, denoted by x ∨ y . {\displaystyle x\vee y.} An element j {\displaystyle j} of A {\displaystyle A}

3696-400: Is equivalent to) that any bounded nonempty subset S {\displaystyle S} of the real numbers has an infimum and a supremum. If S {\displaystyle S} is not bounded below, one often formally writes inf S = − ∞ . {\displaystyle \inf _{}S=-\infty .} If S {\displaystyle S}

3780-743: Is even possible to conclude that sup f ( S ) = f ( sup S ) . {\displaystyle \sup f(S)=f(\sup S).} This may be applied, for instance, to conclude that whenever g {\displaystyle g} is a real (or complex ) valued function with domain Ω ≠ ∅ {\displaystyle \Omega \neq \varnothing } whose sup norm ‖ g ‖ ∞ = def sup x ∈ Ω | g ( x ) | {\displaystyle \|g\|_{\infty }\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\sup _{x\in \Omega }|g(x)|}

3864-659: Is finite, then for every non-negative real number q , {\displaystyle q,} ‖ g ‖ ∞ q   = def   ( sup x ∈ Ω | g ( x ) | ) q = sup x ∈ Ω ( | g ( x ) | q ) {\displaystyle \|g\|_{\infty }^{q}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\sup _{x\in \Omega }|g(x)|\right)^{q}=\sup _{x\in \Omega }\left(|g(x)|^{q}\right)} since

3948-441: Is non-empty then 1 sup S   =   inf 1 S {\displaystyle {\frac {1}{\sup _{}S}}~=~\inf _{}{\frac {1}{S}}} where this equation also holds when sup S = ∞ {\displaystyle \sup _{}S=\infty } if the definition 1 ∞ := 0 {\displaystyle {\frac {1}{\infty }}:=0}

4032-1314: Is not closed under unions then A ∨ B {\displaystyle A\vee B} exists in ( F , ⊆ ) {\displaystyle ({\mathcal {F}},\subseteq )} if and only if there exists a unique ⊆ {\displaystyle \,\subseteq } -smallest J ∈ F {\displaystyle J\in {\mathcal {F}}} such that A ∪ B ⊆ J . {\displaystyle A\cup B\subseteq J.} For example, if F = { { 1 } , { 2 } , { 1 , 2 , 3 } , R } {\displaystyle {\mathcal {F}}=\{\{1\},\{2\},\{1,2,3\},\mathbb {R} \}} then { 1 } ∨ { 2 } = { 1 , 2 , 3 } {\displaystyle \{1\}\vee \{2\}=\{1,2,3\}} whereas if F = { { 1 } , { 2 } , { 1 , 2 , 3 } , { 0 , 1 , 2 } , R } {\displaystyle {\mathcal {F}}=\{\{1\},\{2\},\{1,2,3\},\{0,1,2\},\mathbb {R} \}} then { 1 } ∨ { 2 } {\displaystyle \{1\}\vee \{2\}} does not exist because

4116-473: Is only possible when the given order is not a total one. In a totally ordered set, like the real numbers, the concepts are the same. As an example, let S {\displaystyle S} be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from S {\displaystyle S} together with the set of integers Z {\displaystyle \mathbb {Z} } and

4200-421: Is partially ordered in the usual way (by ⊆ {\displaystyle \,\subseteq } ) then joins are unions and meets are intersections; in symbols, ∨ = ∪  and  ∧ = ∩ {\displaystyle \,\vee \,=\,\cup \,{\text{ and }}\,\wedge \,=\,\cap \,} (where the similarity of these symbols may be used as

4284-408: Is the join (or least upper bound or supremum ) of x  and  y {\displaystyle x{\text{ and }}y} in A {\displaystyle A} if the following two conditions are satisfied: By definition, a binary operation ∧ {\displaystyle \,\wedge \,} on a set A {\displaystyle A}

SECTION 50

#1732855348279

4368-424: Is the union of the subsets when considering the partially ordered set ( P ( X ) , ⊆ ) {\displaystyle (P(X),\subseteq )} , where P {\displaystyle P} is the power set of X {\displaystyle X} and ⊆ {\displaystyle \,\subseteq \,} is subset . Duality (order theory) In

4452-476: Is the converse true: both sets are minimal upper bounds but none is a supremum. The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness . If an ordered set S {\displaystyle S} has the property that every nonempty subset of S {\displaystyle S} having an upper bound also has

4536-747: Is the greatest lower bound of x  and  y {\displaystyle x{\text{ and }}y} with respect to ≤ , {\displaystyle \,\leq ,\,} since z ∧ x = x ∧ z = x ∧ ( x ∧ y ) = ( x ∧ x ) ∧ y = x ∧ y = z {\displaystyle z\wedge x=x\wedge z=x\wedge (x\wedge y)=(x\wedge x)\wedge y=x\wedge y=z} and therefore z ≤ x . {\displaystyle z\leq x.} Similarly, z ≤ y , {\displaystyle z\leq y,} and if w {\displaystyle w}

4620-414: Is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element. However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of

4704-427: Is the least upper bound, a contradiction is immediately deduced because between any two reals x {\displaystyle x} and y {\displaystyle y} (including 2 {\displaystyle {\sqrt {2}}} and p {\displaystyle p} ) there exists some rational r , {\displaystyle r,} which itself would have to be

4788-918: Is the set A + B   :=   { a + b : a ∈ A , b ∈ B } {\displaystyle A+B~:=~\{a+b:a\in A,b\in B\}} consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfies inf ( A + B ) = ( inf A ) + ( inf B ) {\displaystyle \inf(A+B)=(\inf A)+(\inf B)} and sup ( A + B ) = ( sup A ) + ( sup B ) . {\displaystyle \sup(A+B)=(\sup A)+(\sup B).} Product of sets The multiplication of two sets A {\displaystyle A} and B {\displaystyle B} of real numbers

4872-1019: Is used. This equality may alternatively be written as 1 sup s ∈ S s = inf s ∈ S 1 s . {\displaystyle {\frac {1}{\displaystyle \sup _{s\in S}s}}=\inf _{s\in S}{\tfrac {1}{s}}.} Moreover, inf S = 0 {\displaystyle \inf _{}S=0} if and only if sup 1 S = ∞ , {\displaystyle \sup _{}{\tfrac {1}{S}}=\infty ,} where if inf S > 0 , {\displaystyle \inf _{}S>0,} then 1 inf S = sup 1 S . {\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}.} If one denotes by P op {\displaystyle P^{\operatorname {op} }}

4956-816: The least upper bound { 1 } ∨ { 2 } {\displaystyle \{1\}\vee \{2\}} but { 0 , 1 , 2 } ⊈ { 1 , 2 , 3 } {\displaystyle \{0,1,2\}\not \subseteq \{1,2,3\}} and { 1 , 2 , 3 } ⊈ { 0 , 1 , 2 } . {\displaystyle \{1,2,3\}\not \subseteq \{0,1,2\}.} If F = { { 1 } , { 2 } , { 0 , 2 , 3 } , { 0 , 1 , 3 } } {\displaystyle {\mathcal {F}}=\{\{1\},\{2\},\{0,2,3\},\{0,1,3\}\}} then { 1 } ∨ { 2 } {\displaystyle \{1\}\vee \{2\}} does not exist because there

5040-476: The Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic , i.e. if one poset is order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this

5124-405: The mathematical area of order theory , every partially ordered set P gives rise to a dual (or opposite ) partially ordered set which is often denoted by P or P . This dual order P is defined to be the same set, but with the inverse order , i.e. x ≤ y holds in P if and only if y ≤ x holds in P . It is easy to see that this construction, which can be depicted by flipping

SECTION 60

#1732855348279

5208-429: The meet of S {\displaystyle S} is the infimum (greatest lower bound), denoted ⋀ S . {\textstyle \bigwedge S.} In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice . Dually,

5292-643: The weak L p , w {\displaystyle L^{p,w}} space norms (for 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } ), the norm on Lebesgue space L ∞ ( Ω , μ ) , {\displaystyle L^{\infty }(\Omega ,\mu ),} and operator norms . Monotone sequences in S {\displaystyle S} that converge to sup S {\displaystyle \sup S} (or to inf S {\displaystyle \inf S} ) can also be used to help prove many of

5376-512: The formula given below, since addition and multiplication of real numbers are continuous operations. The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout, A , B ⊆ R {\displaystyle A,B\subseteq \mathbb {R} } are sets of real numbers. Sum of sets The Minkowski sum of two sets A {\displaystyle A} and B {\displaystyle B} of real numbers

5460-440: The infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from topology that if f {\displaystyle f} is a continuous function and s 1 , s 2 , … {\displaystyle s_{1},s_{2},\ldots } is a sequence of points in its domain that converges to

5544-478: The infimum of A {\displaystyle A} exists (that is, inf A {\displaystyle \inf A} is a real number) and if p {\displaystyle p} is any real number then p = inf A {\displaystyle p=\inf A} if and only if p {\displaystyle p} is a lower bound and for every ϵ > 0 {\displaystyle \epsilon >0} there

5628-784: The last example, the supremum of a set of rationals is irrational , which means that the rationals are incomplete . One basic property of the supremum is sup { f ( t ) + g ( t ) : t ∈ A }   ≤   sup { f ( t ) : t ∈ A } + sup { g ( t ) : t ∈ A } {\displaystyle \sup\{f(t)+g(t):t\in A\}~\leq ~\sup\{f(t):t\in A\}+\sup\{g(t):t\in A\}} for any functionals f {\displaystyle f} and g . {\displaystyle g.} The supremum of

5712-401: The latter case indeed x {\displaystyle x} is a lower bound of x  and  y , {\displaystyle x{\text{ and }}y,} and since x {\displaystyle x} is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with

5796-430: The least upper bound (if p > 2 {\displaystyle p>{\sqrt {2}}} ) or a member of S {\displaystyle S} greater than p {\displaystyle p} (if p < 2 {\displaystyle p<{\sqrt {2}}} ). Another example is the hyperreals ; there is no least upper bound of the set of positive infinitesimals. There

5880-851: The lower bounds is greater than all the others. However, if there is a meet of x  and  y , {\displaystyle x{\text{ and }}y,} then it is unique, since if both m  and  m ′ {\displaystyle m{\text{ and }}m^{\prime }} are greatest lower bounds of x  and  y , {\displaystyle x{\text{ and }}y,} then m ≤ m ′  and  m ′ ≤ m , {\displaystyle m\leq m^{\prime }{\text{ and }}m^{\prime }\leq m,} and thus m = m ′ . {\displaystyle m=m^{\prime }.} If not all pairs of elements from A {\displaystyle A} have

5964-552: The main ones, one also fixes which operation is considered a meet (the one giving the same order) and which is considered a join (the other one). If ( A , ≤ ) {\displaystyle (A,\leq )} is a partially ordered set , such that each pair of elements in A {\displaystyle A} has a meet, then indeed x ∧ y = x {\displaystyle x\wedge y=x} if and only if x ≤ y , {\displaystyle x\leq y,} since in

6048-1268: The map f : [ 0 , ∞ ) → R {\displaystyle f:[0,\infty )\to \mathbb {R} } defined by f ( x ) = x q {\displaystyle f(x)=x^{q}} is a continuous non-decreasing function whose domain [ 0 , ∞ ) {\displaystyle [0,\infty )} always contains S := { | g ( x ) | : x ∈ Ω } {\displaystyle S:=\{|g(x)|:x\in \Omega \}} and sup S = def ‖ g ‖ ∞ . {\displaystyle \sup S\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\|g\|_{\infty }.} Although this discussion focused on sup , {\displaystyle \sup ,} similar conclusions can be reached for inf {\displaystyle \inf } with appropriate changes (such as requiring that f {\displaystyle f} be non-increasing rather than non-decreasing). Other norms defined in terms of sup {\displaystyle \sup } or inf {\displaystyle \inf } include

6132-545: The notation − A := ( − 1 ) A = { − a : a ∈ A } , {\textstyle -A:=(-1)A=\{-a:a\in A\},} it follows that inf ( − A ) = − sup A  and  sup ( − A ) = − inf A . {\displaystyle \inf(-A)=-\sup A\quad {\text{ and }}\quad \sup(-A)=-\inf A.} Multiplicative inverse of

6216-473: The only ones that are self-dual are the equivalence relations (but the notion of partial order is self-dual). Join and meet All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then

6300-526: The original partial order. Conversely, if ( A , ∧ ) {\displaystyle (A,\wedge )} is a meet-semilattice , and the partial order ≤ {\displaystyle \,\leq \,} is defined as in the universal algebra approach, and z = x ∧ y {\displaystyle z=x\wedge y} for some elements x , y ∈ A , {\displaystyle x,y\in A,} then z {\displaystyle z}

6384-543: The partially-ordered set P {\displaystyle P} with the opposite order relation ; that is, for all x  and  y , {\displaystyle x{\text{ and }}y,} declare: x ≤ y  in  P op  if and only if  x ≥ y  in  P , {\displaystyle x\leq y{\text{ in }}P^{\operatorname {op} }\quad {\text{ if and only if }}\quad x\geq y{\text{ in }}P,} then infimum of

6468-471: The positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part. A lower bound of a subset S {\displaystyle S} of a partially ordered set ( P , ≤ ) {\displaystyle (P,\leq )} is an element y {\displaystyle y} of P {\displaystyle P} such that A lower bound

6552-399: The positive real numbers relative to the real numbers: 0 , {\displaystyle 0,} which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside

6636-511: The set of positive real numbers R + {\displaystyle \mathbb {R} ^{+}} (not including 0 {\displaystyle 0} ) does not have a minimum, because any given element of R + {\displaystyle \mathbb {R} ^{+}} could simply be divided in half resulting in a smaller number that is still in R + . {\displaystyle \mathbb {R} ^{+}.} There is, however, exactly one infimum of

6720-540: The set of positive real numbers R + , {\displaystyle \mathbb {R} ^{+},} ordered by subset inclusion as above. Then clearly both Z {\displaystyle \mathbb {Z} } and R + {\displaystyle \mathbb {R} ^{+}} are greater than all finite sets of natural numbers. Yet, neither is R + {\displaystyle \mathbb {R} ^{+}} smaller than Z {\displaystyle \mathbb {Z} } nor

6804-625: The set of rational numbers. Let S {\displaystyle S} be the set of all rational numbers q {\displaystyle q} such that q 2 < 2. {\displaystyle q^{2}<2.} Then S {\displaystyle S} has an upper bound ( 1000 , {\displaystyle 1000,} for example, or 6 {\displaystyle 6} ) but no least upper bound in Q {\displaystyle \mathbb {Q} } : If we suppose p ∈ Q {\displaystyle p\in \mathbb {Q} }

6888-432: The set, there is another, larger, element. For instance, for any negative real number x , {\displaystyle x,} there is another negative real number x 2 , {\displaystyle {\tfrac {x}{2}},} which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, 0 {\displaystyle 0}

6972-453: The sets { 0 , 1 , 2 }  and  { 1 , 2 , 3 } {\displaystyle \{0,1,2\}{\text{ and }}\{1,2,3\}} are the only upper bounds of { 1 }  and  { 2 } {\displaystyle \{1\}{\text{ and }}\{2\}} in ( F , ⊆ ) {\displaystyle ({\mathcal {F}},\subseteq )} that could possibly be

7056-484: The subset. For non-empty finite subsets, the two approaches yield the same result, and so either may be taken as a definition of meet. In the case where each subset of A {\displaystyle A} has a meet, in fact ( A , ≤ ) {\displaystyle (A,\leq )} is a complete lattice ; for details, see completeness (order theory) . If some power set ℘ ( X ) {\displaystyle \wp (X)}

#278721