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152
Basic Properties of Rough Sets and Rough Membership Function
 Journal of Formalized Mathematics
"... Summary. We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced. ..."
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Summary. We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced.
On the Subcontinua of a Real Line
, 2003
"... In [11] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as]a,b[Q,]a,b[IQ – intervals consisting of rational an ..."
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Cited by 8 (5 self)
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In [11] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as]a,b[Q,]a,b[IQ – intervals consisting of rational and irrational numbers respectively. We show also some basic topological properties of intervals.
Generalized Fashoda meet theorem for unit circle . . .
"... Here we will prove meet theorem for the unit circle and for a square, when 4 points on the boundary are ordered cyclically. Also, the concepts of general rectangle and general circle are defined. ..."
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Cited by 7 (1 self)
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Here we will prove meet theorem for the unit circle and for a square, when 4 points on the boundary are ordered cyclically. Also, the concepts of general rectangle and general circle are defined.
The limit of a real function at a point
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the leftside and the rightside limit of a real function at a point. We prove a few properties of the operations on the proper and improper oneside limits and show that Cauchy and Heine characterizations of the oneside limit are equivalent. ..."
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Summary. We introduce the leftside and the rightside limit of a real function at a point. We prove a few properties of the operations on the proper and improper oneside limits and show that Cauchy and Heine characterizations of the oneside limit are equivalent.
Firstcountable, sequential, and Frechet spaces
 Journal of Formalized Mathematics
, 1998
"... Summary. This article contains a definition of three classes of topological spaces: firstcountable, Frechet, and sequential. Next there are some facts about them, that every firstcountable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of to ..."
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Summary. This article contains a definition of three classes of topological spaces: firstcountable, Frechet, and sequential. Next there are some facts about them, that every firstcountable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not firstcountable. This article is based on [10, pp. 73–81].
Inverse Trigonometric Functions arctan and arccot
 FORMALIZED MATHEMATICS VOL. 16, NO. 2, PAGES 147–158, 2008
, 2008
"... This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot. ..."
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Cited by 7 (3 self)
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This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.
Determinant of Some Matrices of Field Elements
, 2006
"... Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matr ..."
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Cited by 7 (1 self)
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Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.
Brouwer fixed point theorem for disks on the plane.
 Formalized Mathematics,
, 2005
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Inverse Trigonometric Functions Arcsin and Arccos
, 2005
"... Notions of inverse sine and inverse cosine have been introduced. Their basic properties have been proved. ..."
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Cited by 6 (0 self)
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Notions of inverse sine and inverse cosine have been introduced. Their basic properties have been proved.