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Functions and their basic properties
 JOURNAL OF FORMALIZED MATHEMATICS
, 2003
"... The definitions of the mode Function and the graph of a function are introduced. The graph of a function is defined to be identical with the function. The following concepts are also defined: the domain of a function, the range of a function, the identity function, the composition of functions, the ..."
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Cited by 1252 (32 self)
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The definitions of the mode Function and the graph of a function are introduced. The graph of a function is defined to be identical with the function. The following concepts are also defined: the domain of a function, the range of a function, the identity function, the composition of functions, the 11 function, the inverse function, the restriction of a function, the image and the inverse image. Certain basic facts about functions and the notions defined in the article are proved.
Functions from a set to a set
 Journal of Formalized Mathematics
, 1989
"... function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] ..."
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Cited by 1007 (23 self)
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function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] in the new terminology. Also some basic facts about functions of two variables are proved.
Relations and their basic properties
 Journal of Formalized Mathematics
, 1989
"... Summary. We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are ..."
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Cited by 982 (6 self)
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Summary. We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are redefined. Basic facts about the above mentioned notions are presented.
Some Basic Properties of Sets
, 1989
"... this article some basic theorems about singletons, pairs, power sets, unions of families of sets, and the cartesian product of two sets are proved. MML Identifier: ZFMISC1 ..."
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Cited by 721 (21 self)
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this article some basic theorems about singletons, pairs, power sets, unions of families of sets, and the cartesian product of two sets are proved. MML Identifier: ZFMISC1
The fundamental properties of natural numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
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Cited by 641 (72 self)
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Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.html The articles [4], [6], [1], [2], [5], and [3] provide the notation and terminology for this paper. A natural number is an element of N. For simplicity, we use the following convention: x is a real number, k, l, m, n are natural numbers, h, i, j are natural numbers, and X is a subset of R. The following proposition is true (2) 1 For every X such that 0 ∈ X and for every x such that x ∈ X holds x+1 ∈ X and for every k holds k ∈ X. Let n, k be natural numbers. Then n+k is a natural number. Let n, k be natural numbers. Note that n+k is natural. In this article we present several logical schemes. The scheme Ind concerns a unary predicate P, and states that: For every natural number k holdsP[k] provided the parameters satisfy the following conditions: • P[0], and • For every natural number k such thatP[k] holdsP[k+1]. The scheme Nat Ind concerns a unary predicateP, and states that: For every natural number k holdsP[k] provided the following conditions are satisfied: • P[0], and • For every natural number k such thatP[k] holdsP[k+1]. Let n, k be natural numbers. Then n · k is a natural number. Let n, k be natural numbers. Observe that n · k is natural. Next we state several propositions: (18) 2 0 ≤ i. (19) If 0 � = i, then 0 < i. (20) If i ≤ j, then i · h ≤ j · h. 1 The proposition (1) has been removed. 2 The propositions (3)–(17) have been removed.
The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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Cited by 627 (63 self)
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Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on [9] and [10].
Basic Properties of Real Numbers
 Journal of Formalized Mathematics
, 1989
"... this paper. A real number is an element of R ..."
Partial Functions
"... this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules ..."
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Cited by 426 (10 self)
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this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules: x, y, y 1 , y 2 , z, z 1 , z 2 denote sets, P , Q, X , X 0 , X 1 , X 2 , Y , Y 0 , Y 1 , Y 2 , V , Z denote sets, and C, D denote non empty sets. We now state three propositions: (1) If P ` [: X 1
Binary operations
 Journal of Formalized Mathematics
, 1989
"... Summary. In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations:... is commutative,... is associative,... is the unity of..., and... is distributive wrt.... A number of schemes useful in justifying the existence of the operat ..."
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Cited by 330 (6 self)
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Summary. In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations:... is commutative,... is associative,... is the unity of..., and... is distributive wrt.... A number of schemes useful in justifying the existence of the operations are proved. MML Identifier:BINOP_1. WWW:http://mizar.org/JFM/Vol1/binop_1.html The articles [4], [3], [5], [6], [1], and [2] provide the notation and terminology for this paper. Let f be a function and let a, b be sets. The functor f(a, b) yielding a set is defined by: (Def. 1) f(a, b) = f(〈a, b〉). In the sequel A is a set. Let A, B be non empty sets, let C be a set, let f be a function from [:A, B:] into C, let a be an element of A, and let b be an element of B. Then f(a, b) is an element of C. The following proposition is true (2) 1 Let A, B, C be non empty sets and f1, f2 be functions from [:A, B:] into C. Suppose that for every element a of A and for every element b of B holds f1(a, b) = f2(a, b). Then f1 = f2. Let A be a set. A unary operation on A is a function from A into A. A binary operation on A is a
Finite sets
 Journal of Formalized Mathematics
, 1989
"... Summary. The article contains the definition of a finite set based on the notion of finite sequence. Some theorems about properties of finite sets and finite families of sets are proved. ..."
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Cited by 310 (6 self)
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Summary. The article contains the definition of a finite set based on the notion of finite sequence. Some theorems about properties of finite sets and finite families of sets are proved.