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446
Families of sets
 Journal of Formalized Mathematics
, 1989
"... Summary. The article contains definitions of the following concepts: family of sets, family of subsets of a set, the intersection of a family of sets. Functors ∪, ∩, and \ are redefined for families of subsets of a set. Some properties of these notions are presented. ..."
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Cited by 301 (5 self)
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Summary. The article contains definitions of the following concepts: family of sets, family of subsets of a set, the intersection of a family of sets. Functors ∪, ∩, and \ are redefined for families of subsets of a set. Some properties of these notions are presented.
Finite Sequences and Tuples of Elements of a Nonempty Sets
, 1990
"... this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D ..."
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Cited by 291 (7 self)
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this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D
Domains and their Cartesian products
 Journal of Formalized Mathematics
, 1989
"... Summary. The article includes: theorems related to domains, theorems related to Cartesian products presented earlier in various articles and simplified here by substituting domains for sets and omitting the assumption that the sets involved must not be empty. Several schemes and theorems related to ..."
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Cited by 290 (23 self)
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Summary. The article includes: theorems related to domains, theorems related to Cartesian products presented earlier in various articles and simplified here by substituting domains for sets and omitting the assumption that the sets involved must not be empty. Several schemes and theorems related to Fraenkel operator are given. We also redefine subset yielding functions such as the pair of elements of a set and the union of two subsets of a set.
Tuples, projections and Cartesian products
 Journal of Formalized Mathematics
, 1989
"... Summary. The purpose of this article is to define projections of ordered pairs, and to introduce triples and quadruples, and their projections. The theorems in this paper may be roughly divided into two groups: theorems describing basic properties of introduced concepts and theorems related to the r ..."
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Cited by 267 (39 self)
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Summary. The purpose of this article is to define projections of ordered pairs, and to introduce triples and quadruples, and their projections. The theorems in this paper may be roughly divided into two groups: theorems describing basic properties of introduced concepts and theorems related to the regularity, analogous to those proved for ordered pairs by Cz. Byliński [1]. Cartesian products of subsets are redefined as subsets of Cartesian products.
Pigeon hole principle
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the notion of a predicate that states that a function is onetoone at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given ..."
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Cited by 262 (13 self)
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Summary. We introduce the notion of a predicate that states that a function is onetoone at a given element of its domain (i.e. counterimage of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given element of the range of the finite sequence, the substring preceding (and succeeding) the first occurrence of given element of the range. At the end of the article we prove the pigeon hole principle.
Binary operations applied to functions
 Journal of Formalized Mathematics
, 1989
"... Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The ..."
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Cited by 256 (43 self)
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Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a nonempty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.
Basis of Real Linear Space
, 1990
"... this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G ..."
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Cited by 250 (21 self)
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this paper. For simplicity, we follow the rules: x is a set, a, b are real numbers, V is a real linear space, W 1 , W 2 , W 3 are subspaces of V , v, v 1 , v 2 are vectors of V , A, B are subsets of the carrier of V , L, L 1 , L 2 are linear combinations of V , l is a linear combination of A, F , G
The Modification of a Function by a Function and the Iteration of the Composition of a Function
, 1990
"... ..."
The Reflection Theorem
 Journal of Formalized Mathematics
, 1990
"... this paper (and in another Mizar articles) we work in TarskiGrothendieck (TG) theory (see [17]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [15] and some concepts of MK are modeled. The concepts are ..."
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Cited by 228 (50 self)
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this paper (and in another Mizar articles) we work in TarskiGrothendieck (TG) theory (see [17]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [15] and some concepts of MK are modeled. The concepts are: the class On of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of nonempty elements of universe, etc. The reflection theorem states that if A ¸ is an increasing and continuous transfinite sequence of nonempty sets and class A =
Manysorted sets
 Journal of Formalized Mathematics
, 1993
"... Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is def ..."
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Cited by 194 (23 self)
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Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is defined as ∀i∈Ixi ∈ Xi. I was prompted by a remark in a paper by Tarlecki and Wirsing: “Throughout the paper we deal with manysorted sets, functions, relations etc.... We feel free to use any standard settheoretic notation without explicit use of indices ” [6, p. 97]. The aim of this work was to check the feasibility of such approach in Mizar. It works. Let us observe some peculiarities: empty set (i.e. the many sorted set with empty set of indices) belongs to itself (theorem 133), we get two different inclusions X ⊆ Y iff ∀i∈IXi ⊆ Yi and X ⊑ Y iff ∀xx ∈ X ⇒ x ∈ Y equivalent only for sets that yield non empty values. Therefore the care is advised.