WLOG, or the misery of the unordered pair (EWD1223)

1997 ◽  
pp. 33-34
Author(s):  
Edsger W. Dijkstra
Keyword(s):  
Unordered Pair

A Room Design of Order 10

1964 ◽  
Vol 7 (3) ◽  
pp. 377-378 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Positive Integer ◽  
Unordered Pair ◽  
Room Design

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square of side 2n - 1, so that each of the (2n - 1)2 cells of the array is either empty or contains just two distinct objects; each of the 2n objects occurs just once in each row and in each column; and each (unordered) pair of objects occurs in just one cell.

A Room Design of Order 14

1968 ◽  
Vol 11 (2) ◽  
pp. 191-194 ◽  
Author(s):  
C D. O'Shaughnessy
Keyword(s):  
Positive Integer ◽  
Unordered Pair ◽  
Room Design ◽  
The Right ◽  
Square Array ◽  
Being Moved

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of side 2n - 1, such that each of the (2n - 1)2 cells of the array is either empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. A Room design of order 2n is said to be cyclic if the entries in the (i + l) th row are obtained by moving the entries in the i th row one column to the right (with entries in the (2n - l)th column being moved to the first column), and increasing the entries in each occupied cell by l(mod 2n - 1), except that the digit 0 remains unchanged.

A partial model for Quine's “New foundations”

1954 ◽  
Vol 19 (3) ◽  
pp. 197-200 ◽  
Author(s):  
Václav Edvard Beneš
Keyword(s):  
Direct Product ◽  
Set Theory ◽  
Natural Numbers ◽  
Unordered Pair ◽  
Partial Model ◽  
New Foundations ◽  
The Universe ◽  
Ordered Pairs

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model.2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples.We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

An Existence Theorem for Room Squares*

1969 ◽  
Vol 12 (4) ◽  
pp. 493-497 ◽  
Author(s):  
R. C. Mullin ◽  
E. Nemeth
Keyword(s):  
Positive Integer ◽  
Existence Theorem ◽  
Prime Power ◽  
Unordered Pair ◽  
Room Squares ◽  
Square Array

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

scholarly journals Set mappings defined on pairs

1981 ◽  
Vol 30 (3) ◽  
pp. 356-365 ◽  
Author(s):  
N. H. Williams
Keyword(s):  
Chain Condition ◽  
Large Set ◽  
Unordered Pair ◽  
Free Set ◽  
Set Mappings

AbstractA set mapping on pairs over the set S is a function f such that for each unordered pair a of elements of S,f(a) is a subset of S disjoint from a. A subset H of S is said to be free for f if x∉ f({y, z}) for all x, y, z from H. In this paper, we investigate conditions imposed on the range of f which ensure that there is a large set free for f. For example, we show that if f is defined on a set of size K+ + with always |f(a)| <k then f has a free set of size K+ if the range of f satisfies the k-chain condition, or if any two sets in the range of f have an intersection of size less than θ for some θ with θ < K.

scholarly journals Decomposition of the n-Dimensional Lattice-Graph into Hamiltonian Lines

1961 ◽  
Vol 12 (3) ◽  
pp. 123-131 ◽  
Author(s):  
C. ST.J. A. Nash-Williams
Keyword(s):  
Euclidean Space ◽  
Unit Length ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Edge Disjoint ◽  
Unordered Pair ◽  
Spanning Subgraph ◽  
Lattice Graph ◽  
Disjoint Sets

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

scholarly journals Duplication of Room squares

1972 ◽  
Vol 14 (1) ◽  
pp. 75-81 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Unordered Pair ◽  
Room Squares ◽  
Square Array

ARoom squareRof order 2nis a way of arranging 2nobjects (usually 1,2,…, 2n) in a square arrayRof side 2n– 1 so that:(i) every cell of the array is empty or contains two objects;(ii) each unordered pair of objects occurs once inR(iii) every row and column ofRcontains one copy of each object.

Maximal Subsets of a given Set having No Triple in Common with a Steiner Triple System on the set

1969 ◽  
Vol 12 (6) ◽  
pp. 777-778 ◽  
Author(s):  
N. Sauer ◽  
J. Schönheim
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
List Type ◽  
Type Number ◽  
Recent Letter ◽  
Unordered Pair ◽  
Maximal Size ◽  
Finite Set ◽  
Image Position

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:(i)(ii)for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.

Asymptotic Existence of Resolvable Graph Designs

2007 ◽  
Vol 50 (4) ◽  
pp. 504-518 ◽  
Author(s):  
Peter Dukes ◽  
Alan C. H. Ling
Keyword(s):  
Necessary Conditions ◽  
Block Design ◽  
Simple Graph ◽  
Analogous Problem ◽  
Block Graph ◽  
Unordered Pair ◽  
Famous Result ◽  
Graph Design ◽  
Graph Designs ◽  
Vertex Sets

AbstractLet v ≥ k ≥ 1 and λ ≥ 0 be integers. A block design BD(v, k, λ) is a collection of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly λ elements of . More generally, for a fixed simple graph G, a graph design GD(v, G, λ) is a collection of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly λ elements of . A famous result of Wilson says that for a fixed G and λ, there exists a GD(v, G, λ) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v, k, λ), yet for over twenty years the analogous problem for resolvable GD(v, G, λ) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

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