#REDIRECT Subset
If X and Y are sets and every element (mathematics) of X is also an element of Y, then we say or write: * X is a subset of (or is included in) Y; * X ⊆ Y; * Y is a superset of (or includes) X; * Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equal (math) to Y is called proper (or strict). If X is a proper subset of Y, then we write X ⊂ Y. Analogous comments apply to supersets. The binary relation "is a subset of" is called inclusion. == Notational variations == There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" to indicate any subset and uses "⊊" to indicate proper subsets. The newer system uses the symbol "⊆" to indicate any subsets and uses "⊂" to indicate proper subsets. Wikipedia uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets. == Examples == * The set {1, 2} is a proper subset of {1, 2, 3}. * The set of natural numbers is a proper subset of the set of rational numbers. * The set {''x'' : ''x'' is a prime number greater than 2000} is a proper subset of {''x'' : ''x'' is an odd number greater than 1000} * Any set is a subset of itself, but not a proper subset. * The empty set, written ∅, is also a subset of any given set ''X''. (This statement is vacuous truth, see proof below) The empty set is always a proper subset, except of itself. == Properties == PROPOSITION 1: The empty set is a subset of every set. Proof: Given any set A, we wish to prove that ∅ is a subset of A. This involves showing that all elements of ∅ are elements of A. But there are no elements of ∅. For the experienced mathematician, the inference "∅ has no elements, so all elements of ∅ are elements of A" is vacuous truth, but it may be more troublesome for the beginner. Since ∅ has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that ∅ was ''not'' a subset of A, we would have to find an element of ∅ which was not also an element of A. Since there are no elements of ∅, this is impossible and hence ∅ is indeed a subset of A. The following proposition says that inclusion is a partial order. PROPOSITION 2: If ''A'', ''B'' and ''C'' are sets then the following hold: :reflexive relation: ::*''A'' ⊆ ''A'' :antisymmetric relation: ::*''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' if and only if ''A'' = ''B'' :transitive relation: ::*If ''A'' ⊆ ''B'' and ''B'' ⊆ ''C'' then ''A'' ⊆ ''C'' The following proposition says that for any set ''S'' the power set of ''S'' ordered by inclusion is a lattice (order), and hence together with the distributive and complement laws above, show that it is a Boolean algebra. PROPOSITION 3: If ''A'', ''B'' and ''C'' are subsets of a set ''S'' then the following hold: :existence of a greatest element and a greatest element: ::*∅ ⊆ ''A'' ⊆ ''S'' (that ∅ ⊆ ''A'' is Proposition 1 above.) :existence of lattice (order): ::*''A'' ⊆ ''A''∪''B'' ::*If ''A'' ⊆ ''C'' and ''B'' ⊆ ''C'' then ''A''∪''B'' ⊆ ''C'' :existence of lattice (order): ::*''A''∩''B'' ⊆ ''A'' ::*If ''C'' ⊆ ''A'' and ''C'' ⊆ ''B'' then ''C'' ⊆ ''A''∩''B'' The following proposition says that, the statement "''A'' ⊆ ''B'' ", is equivalent to various other statements involving union (set theory), intersection (set theory) and complement (set theory). PROPOSITION 4: For any two sets ''A'' and ''B'', the following are equivalent: :*''A'' ⊆ ''B'' :*''A'' ∩ ''B'' = ''A'' :*''A'' ∪ ''B'' = ''B'' :*''A'' − ''B'' = ∅ :*''B''′ ⊆ ''A''′ The above proposition shows that the relation of set inclusion can be characterized by either of the set operations of union or intersection, which means that the notion of set inclusion is axiomatically superfluous. == Outside of mathematics == "Subset" is commonly used in American English just to mean "type", "sort", "kind". Set theory
==Moved paragraphs from Power set, and question== I moved the last two paragraphs here from :Power set, since they seem more appropriate here. By the way, is it equivalent to say that X ⊂ Y iff X contains no elements that are not members of Y? It seems essentially the same, and is more obviously satisfied by the empty set. - User:Stuart Presnell Yes, that's equivalent. --AxelBoldt ==Merge with proper subset, and superset== Given that this is an encyclopedia and wikipedia:wikipedia is not a dictionary, is any purpose served by having 3 articles, subset, proper subset, and superset? If there's something interesting to say about one of these that doesn't fit in with another, then yes, but I doubt that this is the case. If there are no objections, I'll be combining the articles here and redirecting the others (and maybe the yet nonexistent proper superset too). -- user:Toby Bartels, Sunday, May 19, 2002 good idea! - user:Iwnbap == quick edit == Corrected the inline definitions in the top section. A previous author/editor indicated that X being a subset of Y is the equivalent of 'X includes Y', and that X being a superset of Y is the equivalent of 'X is included in Y'. Actually, these are exactly opposite, and my edit reflects that. (''unsigned comment by anon: 68.43.187.249 00:22, Sep 16, 2004'') ==Any set is a subset of itself, but not a proper subset?== I am a novice, but my sense is that Apostol (pg501 of vol II of Calculus 2nd ed) would disagree with this: "Any set is a subset of itself, but not a proper subset" (''Unsigned comment by User:Swissdude 19:10, Apr 22, 2005'') :Both statements follow immediately from the definitions, which are standard. What does Apostol say on pg501? User:Paul August User_talk:Paul August 04:06, Apr 23, 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Subset:
SubSet
Subset
Subset
Subset-equational_language
Subset-sum_problem
Subsets
Subset_sum
Subset_sum_problem
Subset_sum_problem
SubSet
#REDIRECT Subset
Subset
If X and Y are sets and every element (mathematics) of X is also an element of Y, then we say or write: * X is a subset of (or is included in) Y; * X ⊆ Y; * Y is a superset of (or includes) X; * Y ⊇ X. Every set Y is a subset of itself. A subset of Y which is not equal (math) to Y is called proper (or strict). If X is a proper subset of Y, then we write X ⊂ Y. Analogous comments apply to supersets. The binary relation "is a subset of" is called inclusion. == Notational variations == There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" to indicate any subset and uses "⊊" to indicate proper subsets. The newer system uses the symbol "⊆" to indicate any subsets and uses "⊂" to indicate proper subsets. Wikipedia uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets. == Examples == * The set {1, 2} is a proper subset of {1, 2, 3}. * The set of natural numbers is a proper subset of the set of rational numbers. * The set {''x'' : ''x'' is a prime number greater than 2000} is a proper subset of {''x'' : ''x'' is an odd number greater than 1000} * Any set is a subset of itself, but not a proper subset. * The empty set, written ∅, is also a subset of any given set ''X''. (This statement is vacuous truth, see proof below) The empty set is always a proper subset, except of itself. == Properties == PROPOSITION 1: The empty set is a subset of every set. Proof: Given any set A, we wish to prove that ∅ is a subset of A. This involves showing that all elements of ∅ are elements of A. But there are no elements of ∅. For the experienced mathematician, the inference "∅ has no elements, so all elements of ∅ are elements of A" is vacuous truth, but it may be more troublesome for the beginner. Since ∅ has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that ∅ was ''not'' a subset of A, we would have to find an element of ∅ which was not also an element of A. Since there are no elements of ∅, this is impossible and hence ∅ is indeed a subset of A. The following proposition says that inclusion is a partial order. PROPOSITION 2: If ''A'', ''B'' and ''C'' are sets then the following hold: :reflexive relation: ::*''A'' ⊆ ''A'' :antisymmetric relation: ::*''A'' ⊆ ''B'' and ''B'' ⊆ ''A'' if and only if ''A'' = ''B'' :transitive relation: ::*If ''A'' ⊆ ''B'' and ''B'' ⊆ ''C'' then ''A'' ⊆ ''C'' The following proposition says that for any set ''S'' the power set of ''S'' ordered by inclusion is a lattice (order), and hence together with the distributive and complement laws above, show that it is a Boolean algebra. PROPOSITION 3: If ''A'', ''B'' and ''C'' are subsets of a set ''S'' then the following hold: :existence of a greatest element and a greatest element: ::*∅ ⊆ ''A'' ⊆ ''S'' (that ∅ ⊆ ''A'' is Proposition 1 above.) :existence of lattice (order): ::*''A'' ⊆ ''A''∪''B'' ::*If ''A'' ⊆ ''C'' and ''B'' ⊆ ''C'' then ''A''∪''B'' ⊆ ''C'' :existence of lattice (order): ::*''A''∩''B'' ⊆ ''A'' ::*If ''C'' ⊆ ''A'' and ''C'' ⊆ ''B'' then ''C'' ⊆ ''A''∩''B'' The following proposition says that, the statement "''A'' ⊆ ''B'' ", is equivalent to various other statements involving union (set theory), intersection (set theory) and complement (set theory). PROPOSITION 4: For any two sets ''A'' and ''B'', the following are equivalent: :*''A'' ⊆ ''B'' :*''A'' ∩ ''B'' = ''A'' :*''A'' ∪ ''B'' = ''B'' :*''A'' − ''B'' = ∅ :*''B''′ ⊆ ''A''′ The above proposition shows that the relation of set inclusion can be characterized by either of the set operations of union or intersection, which means that the notion of set inclusion is axiomatically superfluous. == Outside of mathematics == "Subset" is commonly used in American English just to mean "type", "sort", "kind". Set theory
Subset
==Moved paragraphs from Power set, and question== I moved the last two paragraphs here from :Power set, since they seem more appropriate here. By the way, is it equivalent to say that X ⊂ Y iff X contains no elements that are not members of Y? It seems essentially the same, and is more obviously satisfied by the empty set. - User:Stuart Presnell Yes, that's equivalent. --AxelBoldt ==Merge with proper subset, and superset== Given that this is an encyclopedia and wikipedia:wikipedia is not a dictionary, is any purpose served by having 3 articles, subset, proper subset, and superset? If there's something interesting to say about one of these that doesn't fit in with another, then yes, but I doubt that this is the case. If there are no objections, I'll be combining the articles here and redirecting the others (and maybe the yet nonexistent proper superset too). -- user:Toby Bartels, Sunday, May 19, 2002 good idea! - user:Iwnbap == quick edit == Corrected the inline definitions in the top section. A previous author/editor indicated that X being a subset of Y is the equivalent of 'X includes Y', and that X being a superset of Y is the equivalent of 'X is included in Y'. Actually, these are exactly opposite, and my edit reflects that. (''unsigned comment by anon: 68.43.187.249 00:22, Sep 16, 2004'') ==Any set is a subset of itself, but not a proper subset?== I am a novice, but my sense is that Apostol (pg501 of vol II of Calculus 2nd ed) would disagree with this: "Any set is a subset of itself, but not a proper subset" (''Unsigned comment by User:Swissdude 19:10, Apr 22, 2005'') :Both statements follow immediately from the definitions, which are standard. What does Apostol say on pg501? User:Paul August User_talk:Paul August 04:06, Apr 23, 2005 (UTC)
See other meanings of words starting from letter:
S
SB | SC | SD | SE | SF | SG | SH | SI | SJ | SK | SL | SM | SN | SO | SP | SR | SS | ST | SU | SW | SX | SY | SZ |Words begining with Subset:
SubSet
Subset
Subset
Subset-equational_language
Subset-sum_problem
Subsets
Subset_sum
Subset_sum_problem
Subset_sum_problem
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