Как пишется подмножества

{} набор набор элементов A = {3,7,9,14},
B = {9,14,28} | такой, что так что A = { x | x ∈ mathbb {R}, x <0} A⋂B пересечение объекты, принадлежащие множеству A и множеству B A ⋂ B = {9,14} A⋃B союз объекты, принадлежащие множеству A или множеству B A ⋃ B = {3,7,9,14,28} A⊆B подмножество A является подмножеством B. множество A включено в набор B. {9,14,28} ⊆ {9,14,28} A⊂B правильное подмножество / строгое подмножество A является подмножеством B, но A не равно B. {9,14} ⊂ {9,14,28} A⊄B не подмножество множество A не является подмножеством множества B {9,66} ⊄ {9,14,28} A⊇B суперсет A является надмножеством B. множество A включает множество B {9,14,28} ⊇ {9,14,28} A⊃B правильный суперсет / строгий суперсет A является надмножеством B, но B не равно A. {9,14,28} ⊃ {9,14} A⊅B не суперсет множество A не является надмножеством множества B {9,14,28} ⊅ {9,66} 2 А набор мощности все подмножества A   mathcal {P} (А) набор мощности все подмножества A   А = В равенство оба набора имеют одинаковые элементы A = {3,9,14},
B = {3,9,14},
A = B А в дополнять все объекты, не принадлежащие множеству A   А ‘ дополнять все объекты, не принадлежащие множеству A   А Б относительное дополнение объекты, принадлежащие A, а не B A = {3,9,14},
B = {1,2,3},
A B = {9,14} AB относительное дополнение объекты, принадлежащие A, а не B A = {3,9,14},
B = {1,2,3},
A — B = {9,14} A∆B симметричная разница объекты, принадлежащие A или B, но не их пересечение A = {3,9,14},
B = {1,2,3},
A ∆ B = {1,2,9,14} A⊖B симметричная разница объекты, принадлежащие A или B, но не их пересечение A = {3,9,14},
B = {1,2,3},
A ⊖ B = {1,2,9,14} a ∈A элемент,
принадлежит установить членство A = {3,9,14}, 3 ∈ A x ∉A не элемент нет установленного членства A = {3,9,14}, 1 ∉ A ( а , б ) упорядоченная пара сборник из 2-х элементов   A × B декартово произведение множество всех упорядоченных пар из A и B   | A | мощность количество элементов множества A A = {3,9,14}, | A | = 3 #A мощность количество элементов множества A A = {3,9,14}, # A = 3 | вертикальная полоса такой, что А = {х | 3 <х <14} ℵ 0 алеф-нуль бесконечная мощность множества натуральных чисел   ℵ 1 алеф-он мощность множества счетных порядковых чисел   Ø пустой набор Ø = {} A = Ø mathbb {U} универсальный набор набор всех возможных значений   ℕ 0 набор натуральных / целых чисел (с нулем) mathbb {N}0 = {0,1,2,3,4, …} 0 ∈ mathbb {N}0 ℕ 1 набор натуральных / целых чисел (без нуля) mathbb {N}1 = {1,2,3,4,5, …} 6 ∈ mathbb {N}1 ℤ набор целых чисел mathbb {Z} = {…- 3, -2, -1,0,1,2,3, …} -6 ∈ mathbb {Z} ℚ набор рациональных чисел mathbb {Q} = { x | x = a / b , a , b ∈ mathbb {Z}и b ≠ 0} 2/6 ∈ mathbb {Q} ℝ набор реальных чисел mathbb {R} = { x | -∞ < х <∞} 6.343434 ∈ mathbb {R} ℂ набор комплексных чисел mathbb {C} = { z | z = a + bi , -∞ < a <∞, -∞ < b <∞} 6 + 2 i ∈ mathbb {C}
Источник

Как записываются множества и подмножества?

Если каждый элемент множества Б является элементом множества А, то множество Б называется подмножеством А. Обозначается: Б ⊂ А. Пример. Сколько существует подмножеств множества А={1, 2, 3}.

Что является множеством и подмножеством?

В математике изучают не только те или иные множества, но и отношения, взаимосвязи между ними (в частности: равенство множеств, включение). Определение 5: Множество В является подмножеством множества А, если каждый элемент множества В является также элементом множества А.

Как обозначают подмножества?

Подмножества Если множество A состоит из элементов, принадлежащих некоторому другому множеству E, то A называют подмножеством или частью множества E. Для обозначения этого используют специальный символ subset , и запись A subset E означает, что «A является подмножеством E».

Что такое подмножество примеры?

Примеры подмножеств: Множество людей является подмножеством приматов, живущих на Земле. Множество квадратов является подмножеством прямоугольников. Множество полосатых летающих слонов – как пустое множество — является подмножеством чего угодно: приматов, чисел, прямоугольников.

Сколько подмножеств в множестве из n элементов?

Теорема 2.1.1. Число подмножеств конечного множества, состоящего из n элементов, равно 2 n . Доказательство. Множество, состоящее из одного элемента a, имеет два (т.

Как найти разность двух множеств?

Можно пользоваться терминами «уменьшаемое», «вычитаемое». Чтобы составить множество-разность надо записать множество, содержащее все элементы множества-уменьшаемого, которые не содержатся в множестве-вычитаемом, т. е. из множества-вычитаемого изъять все элементы, входящие в множество-вычитаемое.

Что такое подмножество в математике 5 класс?

Например: А= {1;2;3;4;5;6;7}. Числа связанные с данным множеством, называются подмножеством множества А. В= {2;4;6} Обозначается : B A. Читается: множество B является подмножеством множества A.

Что такое подмножество 7 класс?

Подмножество может содержать все элементы множества, а может не содержать ни одного (пустое множество; обозначается знаком ∅). Некоторые элементы множества могут принадлежать одновременно разным подмножествам (пример 5.3). Для наглядной геометрической иллюстрации множеств и отношений между ними используют круги Эйлера.

Что называют пересечением?

Пересечением множеств A и B является множество их общих элементов, т. е. всех элементов, принадлежащих и множеству A, и множеству B. Пересечение множеств обозначается: A ∩ B .

Как доказать что множество ограниченно?

Множество X называется ограниченным сверху, если существует действительное число a такое, что x≤a для всех x∈X. Всякое число, обладающее этим свойством, называется верхней гранью множества X.

Что является пересечением двух множеств?

Пересечением множеств A и B является множество их общих элементов, т. е. всех элементов, принадлежащих и множеству A, и множеству B. Пересечение множеств обозначается: A ∩ B .

Чему равна мощность множества?

мощность множества всех k-элементных подмножеств множества N равна Ckn = n! k!(

Какое множество называют универсальным?

Универса́льное мно́жество — в математике множество, содержащее все объекты и все множества. … В аксиоматике фон Неймана — Бернайса — Гёделя существует универсальный класс — класс всех множеств, но множеством он не является. Класс всех множеств является классом объектов категории Set.

Как определить разность?

Чтобы найти разность, надо от уменьшаемого отнять вычитаемое.

Как найти пересечение двух множеств?

Для того чтобы составить пересечение двух числовых множеств, надо последовательно брать элементы первого множества и проверять, принадлежат ли они второму множеству, те из них, которые принадлежат, и будут составлять пересечение.

Что такое соотношение между множествами?

В математике часто используется для обозначения какой-либо связи между предметами или понятиями термин «отношение». Примеры отношений: отношение равенства между двумя или несколькими переменными, фигурами.

Что понимают подмножеством?

Подмножества Под множеством понимают объединение в одно целое объектов, связанных между собой неким свойством. Термин «множество» в математике не всегда обозначает большое количество предметов, оно может состоять и из одного элемента и вообще не содержать элементов, тогда его называют пустым и обозначают Ж.

Источник

From Wikipedia, the free encyclopedia

Euler diagram showing
A is a subset of B,  A ⊆ B,  and conversely B is a superset of A,  B ⊇ A.

In mathematics, 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 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.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

Definition[edit]

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 ,} is a subset of any set X and a proper subset of any set except itself, the inclusion relation subseteq is a partial order on the set {mathcal {P}}(S) (the power set of S—the set of all subsets of S[1]) defined by {displaystyle Aleq Biff Asubseteq B}. We may also partially order {mathcal {P}}(S) by reverse set inclusion by defining {displaystyle Aleq B{text{ if and only if }}Bsubseteq A.}

When quantified, Asubseteq B is represented as {displaystyle forall xleft(xin Aimplies xin Bright).}[2]

We can prove the statement Asubseteq B by applying a proof technique known as the element argument[3]:

Let sets A and B be given. To prove that {displaystyle Asubseteq B,}

  1. suppose that a is a particular but arbitrarily chosen element of A
  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows {displaystyle cin Aimplies cin B} for an arbitrarily chosen element c. Universal generalisation then implies {displaystyle forall xleft(xin Aimplies xin Bright),} which is equivalent to {displaystyle Asubseteq B,} as stated above.

The set of all subsets of A is called its powerset, and is denoted by {mathcal {P}}(A). The set of all k-subsets of A is denoted by {displaystyle {tbinom {A}{k}}}, in analogue with the notation for binomial coefficients, which count the number of k-subsets of an n-element set. In set theory, the notation {displaystyle [A]^{k}} is also common, especially when k is a transfinite cardinal number.

Properties[edit]

  • A set A is a subset of B if and only if their intersection is equal to A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acap B=A.}
  • A set A is a subset of B if and only if their union is equal to B.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acup B=B.}
  • A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}|Acap B|=|A|.}

⊂ and ⊃ symbols[edit]

Some authors use the symbols subset and supset to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols subseteq and {displaystyle supseteq .}[4] For example, for these authors, it is true of every set A that {displaystyle Asubset A.}

Other authors prefer to use the symbols subset and supset to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols subsetneq and {displaystyle supsetneq .}[5] This usage makes subseteq and subset analogous to the inequality symbols leq and {displaystyle <.} For example, if {displaystyle xleq y,} then x may or may not equal y, but if {displaystyle x<y,} then x definitely does not equal y, and is less than y. Similarly, using the convention that subset is proper subset, if {displaystyle Asubseteq B,} then A may or may not equal B, but if {displaystyle Asubset B,} then A definitely does not equal B.

Examples of subsets[edit]

  • The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions Asubseteq B and Asubsetneq B are true.
  • The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus {displaystyle Dsubseteq E} is true, and {displaystyle Dsubsetneq E} is not true (false).
  • Any set is a subset of itself, but not a proper subset. ({displaystyle Xsubseteq X} is true, and {displaystyle Xsubsetneq X} is false for any set X.)
  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}
  • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.
  • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.

Another example in an Euler diagram:

  • A is a proper subset of B.

    A is a proper subset of B.

  • C is a subset but not a proper subset of B.

    C is a subset but not a proper subset of B.

Other properties of inclusion[edit]

Inclusion is the canonical partial order, in the sense that every partially ordered set {displaystyle (X,preceq )} is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then aleq b if and only if {displaystyle [a]subseteq [b].}

For the power set {displaystyle operatorname {mathcal {P}} (S)} of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of {displaystyle k=|S|} (the cardinality of S) copies of the partial order on {0,1} for which {displaystyle 0<1.} This can be illustrated by enumerating {displaystyle S=left{s_{1},s_{2},ldots ,s_{k}right},}, and associating with each subset {displaystyle Tsubseteq S} (i.e., each element of 2^{S}) the k-tuple from {displaystyle {0,1}^{k},} of which the ith coordinate is 1 if and only if s_{i} is a member of T.

See also[edit]

  • Convex subset
  • Inclusion order
  • Region
  • Subset sum problem
  • Subsumptive containment
  • Total subset

References[edit]

  1. ^ Weisstein, Eric W. «Subset». mathworld.wolfram.com. Retrieved 2020-08-23.
  2. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
  3. ^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.
  4. ^ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157
  5. ^ Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

Bibliography[edit]

  • Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

External links[edit]

  • Media related to Subsets at Wikimedia Commons
  • Weisstein, Eric W. «Subset». MathWorld.
Источник

From Wikipedia, the free encyclopedia

Euler diagram showing
A is a subset of B,  A ⊆ B,  and conversely B is a superset of A,  B ⊇ A.

In mathematics, 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 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.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

Definition[edit]

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 ,} is a subset of any set X and a proper subset of any set except itself, the inclusion relation subseteq is a partial order on the set {mathcal {P}}(S) (the power set of S—the set of all subsets of S[1]) defined by {displaystyle Aleq Biff Asubseteq B}. We may also partially order {mathcal {P}}(S) by reverse set inclusion by defining {displaystyle Aleq B{text{ if and only if }}Bsubseteq A.}

When quantified, Asubseteq B is represented as {displaystyle forall xleft(xin Aimplies xin Bright).}[2]

We can prove the statement Asubseteq B by applying a proof technique known as the element argument[3]:

Let sets A and B be given. To prove that {displaystyle Asubseteq B,}

  1. suppose that a is a particular but arbitrarily chosen element of A
  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows {displaystyle cin Aimplies cin B} for an arbitrarily chosen element c. Universal generalisation then implies {displaystyle forall xleft(xin Aimplies xin Bright),} which is equivalent to {displaystyle Asubseteq B,} as stated above.

The set of all subsets of A is called its powerset, and is denoted by {mathcal {P}}(A). The set of all k-subsets of A is denoted by {displaystyle {tbinom {A}{k}}}, in analogue with the notation for binomial coefficients, which count the number of k-subsets of an n-element set. In set theory, the notation {displaystyle [A]^{k}} is also common, especially when k is a transfinite cardinal number.

Properties[edit]

  • A set A is a subset of B if and only if their intersection is equal to A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acap B=A.}
  • A set A is a subset of B if and only if their union is equal to B.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acup B=B.}
  • A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}|Acap B|=|A|.}

⊂ and ⊃ symbols[edit]

Some authors use the symbols subset and supset to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols subseteq and {displaystyle supseteq .}[4] For example, for these authors, it is true of every set A that {displaystyle Asubset A.}

Other authors prefer to use the symbols subset and supset to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols subsetneq and {displaystyle supsetneq .}[5] This usage makes subseteq and subset analogous to the inequality symbols leq and {displaystyle <.} For example, if {displaystyle xleq y,} then x may or may not equal y, but if {displaystyle x<y,} then x definitely does not equal y, and is less than y. Similarly, using the convention that subset is proper subset, if {displaystyle Asubseteq B,} then A may or may not equal B, but if {displaystyle Asubset B,} then A definitely does not equal B.

Examples of subsets[edit]

  • The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions Asubseteq B and Asubsetneq B are true.
  • The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus {displaystyle Dsubseteq E} is true, and {displaystyle Dsubsetneq E} is not true (false).
  • Any set is a subset of itself, but not a proper subset. ({displaystyle Xsubseteq X} is true, and {displaystyle Xsubsetneq X} is false for any set X.)
  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}
  • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.
  • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.

Another example in an Euler diagram:

  • A is a proper subset of B.

    A is a proper subset of B.

  • C is a subset but not a proper subset of B.

    C is a subset but not a proper subset of B.

Other properties of inclusion[edit]

Inclusion is the canonical partial order, in the sense that every partially ordered set {displaystyle (X,preceq )} is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then aleq b if and only if {displaystyle [a]subseteq [b].}

For the power set {displaystyle operatorname {mathcal {P}} (S)} of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of {displaystyle k=|S|} (the cardinality of S) copies of the partial order on {0,1} for which {displaystyle 0<1.} This can be illustrated by enumerating {displaystyle S=left{s_{1},s_{2},ldots ,s_{k}right},}, and associating with each subset {displaystyle Tsubseteq S} (i.e., each element of 2^{S}) the k-tuple from {displaystyle {0,1}^{k},} of which the ith coordinate is 1 if and only if s_{i} is a member of T.

See also[edit]

  • Convex subset
  • Inclusion order
  • Region
  • Subset sum problem
  • Subsumptive containment
  • Total subset

References[edit]

  1. ^ Weisstein, Eric W. «Subset». mathworld.wolfram.com. Retrieved 2020-08-23.
  2. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
  3. ^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.
  4. ^ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157
  5. ^ Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

Bibliography[edit]

  • Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

External links[edit]

  • Media related to Subsets at Wikimedia Commons
  • Weisstein, Eric W. «Subset». MathWorld.
Источник

Из элементов множества можно составлять различные подмножества.

Например, из множества

X=⊲,∨,⊗

 можно составить такие подмножества:

Подмножеством множествa (X) называется любое множество (Y), каждый элемент которого является элементом множества (X).

⊂

 называют знаком включения.

включение3.png

На рисунке фигура (B) полностью находится в фигуре (A).

На математическом языке записывают:

B⊂A

.

Обрати внимание!

Не путай знак принадлежности

∈

 и знак включения

⊂

!

Правильно писать следует

3∈−1,0,1,2,3,4

(число (3) является элементом множества  

−1,0,1,2,3,4

),

В этом случае в левой части находится число, а не множество.

Источники:

Рис. 1. Подмножество, © ЯКласс.

Источник

From Wikipedia, the free encyclopedia

Euler diagram showing
A is a subset of B,  A ⊆ B,  and conversely B is a superset of A,  B ⊇ A.

In mathematics, 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 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.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

Definition[edit]

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 ,} is a subset of any set X and a proper subset of any set except itself, the inclusion relation subseteq is a partial order on the set {mathcal {P}}(S) (the power set of S—the set of all subsets of S[1]) defined by {displaystyle Aleq Biff Asubseteq B}. We may also partially order {mathcal {P}}(S) by reverse set inclusion by defining {displaystyle Aleq B{text{ if and only if }}Bsubseteq A.}

When quantified, Asubseteq B is represented as {displaystyle forall xleft(xin Aimplies xin Bright).}[2]

We can prove the statement Asubseteq B by applying a proof technique known as the element argument[3]:

Let sets A and B be given. To prove that {displaystyle Asubseteq B,}

  1. suppose that a is a particular but arbitrarily chosen element of A
  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows {displaystyle cin Aimplies cin B} for an arbitrarily chosen element c. Universal generalisation then implies {displaystyle forall xleft(xin Aimplies xin Bright),} which is equivalent to {displaystyle Asubseteq B,} as stated above.

The set of all subsets of A is called its powerset, and is denoted by {mathcal {P}}(A). The set of all k-subsets of A is denoted by {displaystyle {tbinom {A}{k}}}, in analogue with the notation for binomial coefficients, which count the number of k-subsets of an n-element set. In set theory, the notation {displaystyle [A]^{k}} is also common, especially when k is a transfinite cardinal number.

Properties[edit]

  • A set A is a subset of B if and only if their intersection is equal to A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acap B=A.}
  • A set A is a subset of B if and only if their union is equal to B.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acup B=B.}
  • A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}|Acap B|=|A|.}

⊂ and ⊃ symbols[edit]

Some authors use the symbols subset and supset to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols subseteq and {displaystyle supseteq .}[4] For example, for these authors, it is true of every set A that {displaystyle Asubset A.}

Other authors prefer to use the symbols subset and supset to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols subsetneq and {displaystyle supsetneq .}[5] This usage makes subseteq and subset analogous to the inequality symbols leq and {displaystyle <.} For example, if {displaystyle xleq y,} then x may or may not equal y, but if {displaystyle x<y,} then x definitely does not equal y, and is less than y. Similarly, using the convention that subset is proper subset, if {displaystyle Asubseteq B,} then A may or may not equal B, but if {displaystyle Asubset B,} then A definitely does not equal B.

Examples of subsets[edit]

  • The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions Asubseteq B and Asubsetneq B are true.
  • The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus {displaystyle Dsubseteq E} is true, and {displaystyle Dsubsetneq E} is not true (false).
  • Any set is a subset of itself, but not a proper subset. ({displaystyle Xsubseteq X} is true, and {displaystyle Xsubsetneq X} is false for any set X.)
  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}
  • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.
  • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.

Another example in an Euler diagram:

  • A is a proper subset of B.

    A is a proper subset of B.

  • C is a subset but not a proper subset of B.

    C is a subset but not a proper subset of B.

Other properties of inclusion[edit]

Inclusion is the canonical partial order, in the sense that every partially ordered set {displaystyle (X,preceq )} is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then aleq b if and only if {displaystyle [a]subseteq [b].}

For the power set {displaystyle operatorname {mathcal {P}} (S)} of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of {displaystyle k=|S|} (the cardinality of S) copies of the partial order on {0,1} for which {displaystyle 0<1.} This can be illustrated by enumerating {displaystyle S=left{s_{1},s_{2},ldots ,s_{k}right},}, and associating with each subset {displaystyle Tsubseteq S} (i.e., each element of 2^{S}) the k-tuple from {displaystyle {0,1}^{k},} of which the ith coordinate is 1 if and only if s_{i} is a member of T.

See also[edit]

  • Convex subset
  • Inclusion order
  • Region
  • Subset sum problem
  • Subsumptive containment
  • Total subset

References[edit]

  1. ^ Weisstein, Eric W. «Subset». mathworld.wolfram.com. Retrieved 2020-08-23.
  2. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
  3. ^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.
  4. ^ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157
  5. ^ Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

Bibliography[edit]

  • Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

External links[edit]

  • Media related to Subsets at Wikimedia Commons
  • Weisstein, Eric W. «Subset». MathWorld.
Источник

From Wikipedia, the free encyclopedia

Euler diagram showing
A is a subset of B,  A ⊆ B,  and conversely B is a superset of A,  B ⊇ A.

In mathematics, 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 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.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

Definition[edit]

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 ,} is a subset of any set X and a proper subset of any set except itself, the inclusion relation subseteq is a partial order on the set {mathcal {P}}(S) (the power set of S—the set of all subsets of S[1]) defined by {displaystyle Aleq Biff Asubseteq B}. We may also partially order {mathcal {P}}(S) by reverse set inclusion by defining {displaystyle Aleq B{text{ if and only if }}Bsubseteq A.}

When quantified, Asubseteq B is represented as {displaystyle forall xleft(xin Aimplies xin Bright).}[2]

We can prove the statement Asubseteq B by applying a proof technique known as the element argument[3]:

Let sets A and B be given. To prove that {displaystyle Asubseteq B,}

  1. suppose that a is a particular but arbitrarily chosen element of A
  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows {displaystyle cin Aimplies cin B} for an arbitrarily chosen element c. Universal generalisation then implies {displaystyle forall xleft(xin Aimplies xin Bright),} which is equivalent to {displaystyle Asubseteq B,} as stated above.

The set of all subsets of A is called its powerset, and is denoted by {mathcal {P}}(A). The set of all k-subsets of A is denoted by {displaystyle {tbinom {A}{k}}}, in analogue with the notation for binomial coefficients, which count the number of k-subsets of an n-element set. In set theory, the notation {displaystyle [A]^{k}} is also common, especially when k is a transfinite cardinal number.

Properties[edit]

  • A set A is a subset of B if and only if their intersection is equal to A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acap B=A.}
  • A set A is a subset of B if and only if their union is equal to B.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}Acup B=B.}
  • A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A.
Formally:
{displaystyle Asubseteq B{text{ if and only if }}|Acap B|=|A|.}

⊂ and ⊃ symbols[edit]

Some authors use the symbols subset and supset to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols subseteq and {displaystyle supseteq .}[4] For example, for these authors, it is true of every set A that {displaystyle Asubset A.}

Other authors prefer to use the symbols subset and supset to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols subsetneq and {displaystyle supsetneq .}[5] This usage makes subseteq and subset analogous to the inequality symbols leq and {displaystyle <.} For example, if {displaystyle xleq y,} then x may or may not equal y, but if {displaystyle x<y,} then x definitely does not equal y, and is less than y. Similarly, using the convention that subset is proper subset, if {displaystyle Asubseteq B,} then A may or may not equal B, but if {displaystyle Asubset B,} then A definitely does not equal B.

Examples of subsets[edit]

  • The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions Asubseteq B and Asubsetneq B are true.
  • The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus {displaystyle Dsubseteq E} is true, and {displaystyle Dsubsetneq E} is not true (false).
  • Any set is a subset of itself, but not a proper subset. ({displaystyle Xsubseteq X} is true, and {displaystyle Xsubsetneq X} is false for any set X.)
  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}
  • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.
  • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.

Another example in an Euler diagram:

  • A is a proper subset of B.

    A is a proper subset of B.

  • C is a subset but not a proper subset of B.

    C is a subset but not a proper subset of B.

Other properties of inclusion[edit]

Inclusion is the canonical partial order, in the sense that every partially ordered set {displaystyle (X,preceq )} is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then aleq b if and only if {displaystyle [a]subseteq [b].}

For the power set {displaystyle operatorname {mathcal {P}} (S)} of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of {displaystyle k=|S|} (the cardinality of S) copies of the partial order on {0,1} for which {displaystyle 0<1.} This can be illustrated by enumerating {displaystyle S=left{s_{1},s_{2},ldots ,s_{k}right},}, and associating with each subset {displaystyle Tsubseteq S} (i.e., each element of 2^{S}) the k-tuple from {displaystyle {0,1}^{k},} of which the ith coordinate is 1 if and only if s_{i} is a member of T.

See also[edit]

  • Convex subset
  • Inclusion order
  • Region
  • Subset sum problem
  • Subsumptive containment
  • Total subset

References[edit]

  1. ^ Weisstein, Eric W. «Subset». mathworld.wolfram.com. Retrieved 2020-08-23.
  2. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
  3. ^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.
  4. ^ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157
  5. ^ Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

Bibliography[edit]

  • Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

External links[edit]

  • Media related to Subsets at Wikimedia Commons
  • Weisstein, Eric W. «Subset». MathWorld.
Источник

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