Citation Context

...l transformations related to terms constructed with the Frænkel Operator. MML Identifier:FRAENKEL. WWW:http://mizar.org/JFM/Vol2/fraenkel.html The articles [6], [3], [8], [9], [4], [7], [1], [2], and =-=[5]-=- provide the notation and terminology for this paper. In this paper B is a non empty set and A, C, X are sets. In this article we present several logical schemes. The scheme Fraenkel5’ deals with a no...

Citation Context

...binary operations on real numbers. MML Identifier: RVSUM_1. WWW: http://mizar.org/JFM/Vol2/rvsum_1.html The articles [17], [21], [8], [2], [18], [12], [1], [19], [22], [5], [7], [6], [4], [15], [14], =-=[16]-=-, [3], [10], [20], [13], and [9] provide the notation and terminology for this paper. For simplicity, we adopt the following rules: i, j, k denote natural numbers, x, r, r1, r2, r3 denote elements of ...

Citation Context

...tion and propositions for inverses are introduced. MML Identifier:FINSEQOP. WWW:http://mizar.org/JFM/Vol2/finseqop.html The articles [11], [7], [12], [13], [4], [1], [6], [5], [3], [2], [9], [8], and =-=[10]-=- provide the notation and terminology for this paper. For simplicity, we follow the rules: x, y denote sets, C, C ′ , D, D ′ , E denote non empty sets, c denotes an element of C, c ′ denotes an elemen...

Citation Context

...ns that have a unity or for non-empty finite sequences. MML Identifier:FINSOP_1. WWW:http://mizar.org/JFM/Vol2/finsop_1.html The articles [11], [15], [12], [3], [6], [2], [14], [16], [17], [4], [13], =-=[10]-=-, [5], [1], [9], and [7] provide the notation and terminology for this paper. For simplicity, we use the following convention: D is a non empty set, d, d1, d2, d3 are elements of D, F, G, H are finite...

Citation Context

...CS Volume 2, Released 1990, Published 2003 Inst. of Computer Science, Univ. of Białystok Semigroup Operations on Finite Subsets Czesław Byliński Warsaw University Białystok Summary. A continuation of =-=[10]-=-. The propositions and theorems proved in [10] are extended to finite sequences. Several additional theorems related to semigroup operations of functions not included in [10] are proved. The special n...

Citation Context

... equivalence between two notions of degrees. MML Identifier: SGRAPH1. WWW: http://mizar.org/JFM/Vol6/sgraph1.html The articles [12], [7], [15], [13], [2], [1], [4], [5], [6], [3], [9], [8], [14], and =-=[11]-=- provide the notation and terminology for this paper. 1. PRELIMINARIES Let m, n be natural numbers. The functor [m,n]N yields a subset of N and is defined by: (Def. 1) [m,n]N = {i;i ranges over natura...