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44-531: [REDACTED] Look up maximal in Wiktionary, the free dictionary. Maximal may refer to: Maximal element , a mathematical definition Maximal set Maximal ( Transformers ) , a faction of Transformers Maximalism , an artistic style Maxim (magazine) , a men's magazine marketed as Maximal in several countries See also [ edit ] Minimal (disambiguation) Topics referred to by

88-566: A preordered set and let S ⊆ P . {\displaystyle S\subseteq P.} A maximal element of S {\displaystyle S} with respect to ≤ {\displaystyle \,\leq \,} is an element m ∈ S {\displaystyle m\in S} such that Similarly, a minimal element of S {\displaystyle S} with respect to ≤ {\displaystyle \,\leq \,}

132-411: A directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element, and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 above . Similar conclusions are true for minimal elements. Further introductory information

176-434: A finite ordered set P {\displaystyle P} is equal to the smallest lower set containing all maximal elements of L . {\displaystyle L.} Inclusion (set theory) In mathematics, a set A is a subset of a set B if all elements of A are also elements of B ; B is then a superset of A . It is possible for A and B to be equal; if they are unequal, then A

220-434: A greatest element; see example 3. If P {\displaystyle P} satisfies the ascending chain condition , a subset S {\displaystyle S} of P {\displaystyle P} has a greatest element if, and only if , it has one maximal element. When the restriction of ≤ {\displaystyle \,\leq \,} to S {\displaystyle S}

264-607: A partially ordered set P {\displaystyle P} is said to be cofinal if for every x ∈ P {\displaystyle x\in P} there exists some y ∈ Q {\displaystyle y\in Q} such that x ≤ y . {\displaystyle x\leq y.} Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements. A subset L {\displaystyle L} of

308-510: A partially ordered set P {\displaystyle P} is said to be a lower set of P {\displaystyle P} if it is downward closed: if y ∈ L {\displaystyle y\in L} and x ≤ y {\displaystyle x\leq y} then x ∈ L . {\displaystyle x\in L.} Every lower set L {\displaystyle L} of

352-476: A preference preorder would be that of most preferred choice. That is, some x ∈ B {\displaystyle x\in B} with y ∈ B {\displaystyle y\in B} implies y ≺ x . {\displaystyle y\prec x.} An obvious application is to the definition of demand correspondence. Let P {\displaystyle P} be

396-472: A quantity of consumption specified for each existing commodity in the economy. Preferences of a consumer are usually represented by a total preorder ⪯ {\displaystyle \preceq } so that x , y ∈ X {\displaystyle x,y\in X} and x ⪯ y {\displaystyle x\preceq y} reads: x {\displaystyle x}

440-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

484-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

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528-734: Is a proper subset of B . The relationship of one set being a subset of another is called inclusion (or sometimes containment ). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B . A k -subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove

572-637: Is a total order ( S = { 1 , 2 , 4 } {\displaystyle S=\{1,2,4\}} in the topmost picture is an example), then the notions of maximal element and greatest element coincide. This is not a necessary condition: whenever S {\displaystyle S} has a greatest element, the notions coincide, too, as stated above. If the notions of maximal element and greatest element coincide on every two-element subset S {\displaystyle S} of P . {\displaystyle P.} then ≤ {\displaystyle \,\leq \,}

616-704: Is a correspondence Γ : P × R + → X {\displaystyle \Gamma \colon P\times \mathbb {R} _{+}\rightarrow X} mapping any price system and any level of income into a subset Γ ( p , m ) = { x ∈ X   :   p ( x ) ≤ m } . {\displaystyle \Gamma (p,m)=\{x\in X~:~p(x)\leq m\}.} The demand correspondence maps any price p {\displaystyle p} and any level of income m {\displaystyle m} into

660-479: Is a maximal element and s ∈ S , {\displaystyle s\in S,} then it remains possible that neither s ≤ m {\displaystyle s\leq m} nor m ≤ s . {\displaystyle m\leq s.} This leaves open the possibility that there exist more than one maximal elements. For a partially ordered set ( P , ≤ ) , {\displaystyle (P,\leq ),}

704-600: Is a maximal element of }}\Gamma (p,m)\right\}.} It is called demand correspondence because the theory predicts that for p {\displaystyle p} and m {\displaystyle m} given, the rational choice of a consumer x ∗ {\displaystyle x^{*}} will be some element x ∗ ∈ D ( p , m ) . {\displaystyle x^{*}\in D(p,m).} A subset Q {\displaystyle Q} of

748-562: Is a total order on P . {\displaystyle P.} Dual to greatest is the notion of least element that relates to minimal in the same way as greatest to maximal . In a totally ordered set , the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any partially ordered set, but also to their order theoretic generalization via directed sets . In

792-822: Is an element m ∈ S {\displaystyle m\in S} such that Equivalently, m ∈ S {\displaystyle m\in S} is a minimal element of S {\displaystyle S} with respect to ≤ {\displaystyle \,\leq \,} if and only if m {\displaystyle m} is a maximal element of S {\displaystyle S} with respect to ≥ , {\displaystyle \,\geq ,\,} where by definition, q ≥ p {\displaystyle q\geq p} if and only if p ≤ q {\displaystyle p\leq q} (for all p , q ∈ P {\displaystyle p,q\in P} ). If

836-719: Is at most as preferred as y {\displaystyle y} . When x ⪯ y {\displaystyle x\preceq y} and y ⪯ x {\displaystyle y\preceq x} it is interpreted that the consumer is indifferent between x {\displaystyle x} and y {\displaystyle y} but is no reason to conclude that x = y . {\displaystyle x=y.} preference relations are never assumed to be antisymmetric. In this context, for any B ⊆ X , {\displaystyle B\subseteq X,} an element x ∈ B {\displaystyle x\in B}

880-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

924-451: Is different from Wikidata All article disambiguation pages All disambiguation pages Maximal element The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S {\displaystyle S} of a preordered set is an element of S {\displaystyle S} which

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968-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

1012-528: Is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem , the Kirszbraun theorem , Tychonoff's theorem , the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field . Let ( P , ≤ ) {\displaystyle (P,\leq )} be

1056-530: Is found in the article on order theory . In economics, one may relax the axiom of antisymmetry, using preorders (generally total preorders ) instead of partial orders; the notion analogous to maximal element is very similar, but different terminology is used, as detailed below. In consumer theory the consumption space is some set X {\displaystyle X} , usually the positive orthant of some vector space so that each x ∈ X {\displaystyle x\in X} represents

1100-448: Is greater than every other element of S . {\displaystyle S.} A subset may have at most one greatest element. The greatest element of S , {\displaystyle S,} if it exists, is also a maximal element of S , {\displaystyle S,} and the only one. By contraposition , if S {\displaystyle S} has several maximal elements, it cannot have

1144-407: Is greater than or equal to any other element of S , {\displaystyle S,} and the minimum of S {\displaystyle S} is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets ,

1188-561: Is not unique for y ⪯ x {\displaystyle y\preceq x} does not preclude the possibility that x ⪯ y {\displaystyle x\preceq y} (while y ⪯ x {\displaystyle y\preceq x} and x ⪯ y {\displaystyle x\preceq y} do not imply x = y {\displaystyle x=y} but simply indifference x ∼ y {\displaystyle x\sim y} ). The notion of greatest element for

1232-400: Is obtained by using ≥ {\displaystyle \,\geq \,} in place of ≤ . {\displaystyle \,\leq .} Maximal elements need not exist. In general ≤ {\displaystyle \,\leq \,} is only a partial order on S . {\displaystyle S.} If m {\displaystyle m}

1276-567: Is said to be a maximal element if y ∈ B {\displaystyle y\in B} implies y ⪯ x {\displaystyle y\preceq x} where it is interpreted as a consumption bundle that is not dominated by any other bundle in the sense that x ≺ y , {\displaystyle x\prec y,} that is x ⪯ y {\displaystyle x\preceq y} and not y ⪯ x . {\displaystyle y\preceq x.} It should be remarked that

1320-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

1364-526: The irreflexive kernel of ≤ {\displaystyle \,\leq \,} is denoted as < {\displaystyle \,<\,} and is defined by x < y {\displaystyle x<y} if x ≤ y {\displaystyle x\leq y} and x ≠ y . {\displaystyle x\neq y.} For arbitrary members x , y ∈ P , {\displaystyle x,y\in P,} exactly one of

Maximal - Misplaced Pages Continue

1408-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

1452-475: The class of functionals on X {\displaystyle X} . An element p ∈ P {\displaystyle p\in P} is called a price functional or price system and maps every consumption bundle x ∈ X {\displaystyle x\in X} into its market value p ( x ) ∈ R + {\displaystyle p(x)\in \mathbb {R} _{+}} . The budget correspondence

1496-552: The element { d , o } is minimal as it contains no sets in the collection, the element { g , o , a , d } is maximal as there are no sets in the collection which contain it, the element { d , o , g } is neither, and the element { o , a , f } is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for S . {\displaystyle S.} Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma

1540-468: The following cases applies: Given a subset S ⊆ P {\displaystyle S\subseteq P} and some x ∈ S , {\displaystyle x\in S,} Thus the definition of a greatest element is stronger than that of a maximal element. Equivalently, a greatest element of a subset S {\displaystyle S} can be defined as an element of S {\displaystyle S} that

1584-415: The formal definition looks very much like that of a greatest element for an ordered set. However, when ⪯ {\displaystyle \preceq } is only a preorder, an element x {\displaystyle x} with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element x ∈ B {\displaystyle x\in B}

1628-450: The notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an example, in the collection S := { { d , o } , { d , o , g } , { g , o , a , d } , { o , a , f } } {\displaystyle S:=\left\{\{d,o\},\{d,o,g\},\{g,o,a,d\},\{o,a,f\}\right\}} ordered by containment ,

1672-1000: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

1716-926: The preordered set ( P , ≤ ) {\displaystyle (P,\leq )} also happens to be a partially ordered set (or more generally, if the restriction ( S , ≤ ) {\displaystyle (S,\leq )} is a partially ordered set) then m ∈ S {\displaystyle m\in S} is a maximal element of S {\displaystyle S} if and only if S {\displaystyle S} contains no element strictly greater than m ; {\displaystyle m;} explicitly, this means that there does not exist any element s ∈ S {\displaystyle s\in S} such that m ≤ s {\displaystyle m\leq s} and m ≠ s . {\displaystyle m\neq s.} The characterization for minimal elements

1760-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

1804-411: The same term [REDACTED] This disambiguation page lists articles associated with the title Maximal . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Maximal&oldid=1249011241 " Category : Disambiguation pages Hidden categories: Short description

Maximal - Misplaced Pages Continue

1848-495: The set of ⪯ {\displaystyle \preceq } -maximal elements of Γ ( p , m ) {\displaystyle \Gamma (p,m)} . D ( p , m ) = { x ∈ X   :   x  is a maximal element of  Γ ( p , m ) } . {\displaystyle D(p,m)=\left\{x\in X~:~x{\text{

1892-917: The statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as a consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

1936-523: The subset S {\displaystyle S} is not specified then it should be assumed that S := P . {\displaystyle S:=P.} Explicitly, a maximal element (respectively, minimal element ) of ( P , ≤ ) {\displaystyle (P,\leq )} is a maximal (resp. minimal) element of S := P {\displaystyle S:=P} with respect to ≤ . {\displaystyle \,\leq .} If

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