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492
Real Function Continuity
, 2002
"... this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R ..."
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Cited by 47 (8 self)
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this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R
Introduction to categories and functors
 Journal of Formalized Mathematics
, 1989
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Categories of Groups
, 2000
"... this paper. In this paper x, y denote sets, D denotes a non empty set, and U 1 denotes a universal class. The following propositions are true: (2) ..."
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Cited by 39 (4 self)
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this paper. In this paper x, y denote sets, D denotes a non empty set, and U 1 denotes a universal class. The following propositions are true: (2)
Topological Spaces
"... this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx; ..."
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Cited by 37 (0 self)
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this paper. The following propositions are true: (1) Let A, B be non empty sets and R 1 , R 2 be relations between A and B. Suppose that for every element x of A and for every element y of B holds hhx; yii 2 R 1 iff hhx;
Transpose Matrices and Groups of Permutations
, 2003
"... Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. ..."
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Cited by 34 (0 self)
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Some facts concerning matrices with dimension 2 × 2 are shown. Upper and lower triangular matrices, and operation of deleting rows and columns in a matrix are introduced. Besides, we deal with sets of permutations and the fact that all permutations of finite set constitute a finite group is proved. Some proofs are based on [11] and [14].
Homomorphisms and isomorphisms of groups. Quotient group
 Journal of Formalized Mathematics
, 1991
"... Summary. Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following [9]. ..."
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Summary. Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following [9].
Subcategories and products of categories
 Journal of Formalized Mathematics
, 1990
"... inclusion functor is the injection (inclusion) map E ֒ → which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that HomE(a,b) = HomC(b,b) for any objects a,b of ..."
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inclusion functor is the injection (inclusion) map E ֒ → which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that HomE(a,b) = HomC(b,b) for any objects a,b of E, are defined. A subcategory E of C is full when the inclusion functor E ֒ → is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories B and C is constructed in the usual way. Moreover, some simple facts on bi f unctors (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors (complex functors and product functors).