Problem 1(a). The domain of f is N × N, so elements of the domain of f are ordered pairs 〈a, b〉 where a, b ∈ N. The following equivalences hold for every ordered pair 〈a, b〉 ∈ N× N. 〈a, b〉 ∈ f[E] ⇐⇒ f(〈a, b〉) ∈ E ⇐⇒ a ∈ E The first equivalence is an immediate consequence of the definition of the inverse image f[E]. The second equivalence follows from the definition of function f . We thus have ...

We consider the class Pn of labeled posets on n elements which avoid certain three-element induced subposets. We show that the number of posets in Pn is (n+1) by exploiting a bijection between Pn and the set of regions of the arrangement of hyperplanes in R of the form xi&xj=0 or 1 for 1 i< j n. It also follows that the number of posets in Pn with i pairs (a, b) such that a<b is equal to the nu...

A transitive triple, (a, b, c), is defined to be the set {(a, b), (b, c), (a, c)} of ordered pairs. A directed triple system of order v, DTS(v), is a pair (D, β), where D is a set of v points and β is a collection of transitive triples of pairwise distinct points of D such that any ordered pair of distinct points of D is contained in precisely one transitive triple of β. An antiautomorphism of ...

Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a + b: a ∈ A, b ∈ B, and P (a, b) 6= 0}| where P (x, y) ∈ F[x, y].

1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ?) where G is a nonempty set and ? is a binary operation on G satisfying: (G1) a ? (b ? c) = (a ? b) ? c, ∀a, b, c ∈ G. A semigroup G is a monoid if it also satisfies: (G2) G has an element e (sometimes denoted by 1G, called the identity of G), such that e ? a = a ? e = a, ∀a ∈ G. A monoid is a group if (G3) below is satisfied...

DOI: 10.1007/s11424-015-2240-3 Received: 8 November 2012 / Revised: 1 July 2014 c ©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2015 Abstract Let k be a positive integer. For any positive integer x = ∑∞ i=0 xi2 , where xi = 0, 1, we define the weight w(x) of x by w(x) := ∑∞ i=0 xi. For any integer t with 0 < t < 2 k − 1, let St := {(a, b) ∈ Z2|a+ b ≡ t (mod 2 − 1), w(a)+w(b)...

A directed BIBD with parameters (v,b,r,k,λ*) is a BIBD with parameters (v, b, r, k, 2λ*) in which each ordered pair of varieties occurs together in exactly λ* blocks. It is shown that λ*v(v 1) = 0 (mod 3) is a necessary and sufficient condition for the existence of a directed (v, b, r, k, λ*) BIBD with k = 3. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J and Skill...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

BACKGROUND Pretravel medication and vaccination recommendations and receipt were compared between primary care providers (PCPs) without special training and clinical pharmacists specializing in pretravel health. METHODS A retrospective chart review of patients seen for pretravel health services in a pharmacist-run travel clinic (PTC) compared to PCPs at a University Student Health Center. Vac...