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12
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.
Lattice of subgroups of a group. Frattini subgroup
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in [21]) and multiplication of subgroups which result ..."
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Cited by 28 (5 self)
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Summary. We define the notion of a subgroup generated by a set of element of a group and two closely connected notions. Namely lattice of subgroups and Frattini subgroup. The operations in the lattice are the intersection of subgroups (introduced in [21]) and multiplication of subgroups which result is defined as a subgroup generated by a sum of carriers of the two subgroups. In order to define Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and nongenerating element of the group (see [9, page 30]). Frattini subgroup is defined as in [9] as an intersection of all maximal subgroups. We show that an element of the group belongs to Frattini subgroup of the group if and only if it is a nongenerating element. We also prove theorems that should be proved in [1] but are not.
Associated Matrix of Linear Map
, 2002
"... this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D ..."
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Cited by 17 (2 self)
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this paper. 1. PRELIMINARIES Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then p i is an element of D
Basic Properties of the Rank of Matrices over a Field
, 2007
"... In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entrie ..."
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Cited by 7 (2 self)
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In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m × n matrix A by the condition: A has the rank r if and only if, there is a r × r submatrix of A with a nonzero determinant, and for every k × k submatrix of A with a nonzero determinant we have k ≤ r. At the end, I prove that the rank defined by the size of the biggest submatrix with a nonzero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.
The rank+nullity theorem
 Formalized Mathematics
"... Summary. The rank+nullity theorem states that, if T is a linear transformation from a finitedimensional vector space V to a finitedimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for ..."
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Cited by 6 (1 self)
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Summary. The rank+nullity theorem states that, if T is a linear transformation from a finitedimensional vector space V to a finitedimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to B − A, and that T is onetoone on B − A.
Pocklington’s Theorem and Bertrand’s Postulate
"... Summary. The first four sections of this article include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington’s theorem (see [19]). The last section presents the formalization of Bertrand’s postulate closely following the book [1], ..."
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Cited by 2 (0 self)
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Summary. The first four sections of this article include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington’s theorem (see [19]). The last section presents the formalization of Bertrand’s postulate closely following the book [1], pp. 7–9.
Linear Transformations of Euclidean Topological Spaces
"... Summary. We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomor ..."
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Cited by 1 (1 self)
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Summary. We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.
The Perfect Number Theorem and Wilson’s Theorem
"... Summary. This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The ..."
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Summary. This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson’s theorem (that n is prime iff n> 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler’s sum of divisors function φ, proves that φ is multiplicative and that ∑ φ(k) = n. kn
The JordanHölder Theorem
"... Summary. The goal of this article is to formalize the JordanHölder theorem in the context of group with operators as in the book [5]. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library ..."
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Summary. The goal of this article is to formalize the JordanHölder theorem in the context of group with operators as in the book [5]. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library. Next, the article formalizes the Zassenhaus butterfly lemma and the Schreier refinement theorem, and defines the composition series.