Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 99-101). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 99-101). Also available on the Internet.

Inaug.-Diss.- Bonn. / Includes bibliographical references (p. 58).

Inaug.-Diss.- Bonn. / Includes bibliographical references (p. 58).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 63).

Sidman and Tailby (1982) established procedures to analyze the nature of stimulus to stimulus relations established by conditional discriminations. Their research describes specific behavioral tests to determine the establishment of properties that define the relations of equivalence. An equivalence relation requires the demonstration of three conditional relations: reflexivity, symmetry, and transitivity. The equivalence stimulus paradigm provides a method to account for novel responding. The research suggests that equivalence relations provide a more efficient and effective approach to the assessment, analysis, and instruction of skills. The present research examined the effectiveness of the formation of an equivalence class in teaching young children coin value relations. The second aspect of the study was to determine if there was a relationship between equivalence class formation and generalization of the skills established to other settings. Five children, 4- and 5-years old, were selected to participate in the study based on their lack of skills in the area of coin values and purchasing an item with dimes or quarters equaling fifth cents. The experimental task was presented on a Macintosh computer with HyperCard programming. The experimental stimuli consisted of pictures of dimes, quarters, and Hershey candy bars presented in match-to-sample procedures. Two conditional discriminations were taught (if A then B and if B then C.). The formation of an equivalence class was evaluated by if C then A. Generalization across settings was tested after the formation of an equivalence class by having the children purchase a Hershey candy bar with dimes at a play store. A multiple baseline experimentaldesign was used to demonstrate a functional relationship between the formation of an equivalence class and generalization of skills across settings. The present research provides supportive evidence that coin value relations can be taught to young children using equivalence procedures. The study also demonstrated generalization of novel, untaught stimuli across settings, after the formation of an equivalence class. A posttest on generalization across settings was conducted 3 months after the study. Long-term stability of equivalence relations was demonstrated by three of the subjects. / Department of Special Education

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

An equivalence class exists if the stimuli that comprise the class are related by the properties of reflexivity, symmetry, and transitivity. Through these properties, new behavior that has not been taught emerges. For example, when taught to match Set A stimuli to Set B stimuli and to match Set A stimuli to Set C stimuli, if equivalence classes have formed, subjects will (with no explicit instruction) match Set B stimuli to Set C and Set C stimuli to Set B stimuli. Although equivalence classes have been studied extensively, few studies have considered the application of this technology to educational concerns. The purpose of this study was (a) to determine if phonics skills could effectively and efficiently be taught to young children through the formation of equivalence classes and (b) to investigate the generality of those acquired skills. Using a conditional discrimination procedure, children were taught to match printed letters to dictated phonetic sounds and to match the initial sound of pictured items to dictated phonetic sounds. Test results indicated that equivalence classes had emerged and that generalization did occur. The children could match the initial sound of pictured items to printed letters and vice versa and could name letter sounds and initial sounds of pictured items. During generality testing, each child could identify the initial sound of several novel pictured items and could sound out the letters within the words. However, reading did not occur. Only 1 of 5 children could blend the sounds of letters into recognizable words. A significant difficulty encountered throughout the study was maintaining the children's motivation, possibly due to the children's inexperience in attending to academic tasks. This study did, however, demonstrate that the formation of equivalence classes is an effective and efficient method for teaching phonics and that the formation of generalized equivalence classes is effective in extending those taught relations to novel stimuli. / Department of Special Education

Let X be a Polish space and Q a measurable partition of X into Gδ equivalence classes. In 1978, S. M. Srivastava proved the existence of a Borel cross section for Q. He asked whether more can be concluded in case each equivalence class is uncountable. This question is answered here in the affirmative. The main result of the author is a proof that shows the existence of a Castaing Representation for Q.

The purpose of this monograph is to make a detailed study about studying of relations. So, we will provide some definitions. Developing this Themes like binary relations, equivalence relations and order relations. At the same we will show ways to represent relations highlighting the representation in the form of graphs. And we will finish the job with Topological Ordering in completing some tasks of a project. / O objetivo desta monografia é fazer um estudo detalhado sobre relações. Para tanto, forneceremos algumas definições. Para o desenvolvimento deste foram abordados temas como relações binárias, relações de equivalência e relações de ordem. No desenvolvimento deste veremos formas de representar as relações dando destaque a representação na forma de Grafos. E finalizamos o trabalho com a Ordenação Topológica na conclusão de algumas tarefas de um projeto.