1.1GenForm062610

1.1GenForm062610 - 1 CHAPTER 1 INTRODUCTION TO BRT 1.1....

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1 CHAPTER 1 INTRODUCTION TO BRT 1.1. General Formulation. 1.2. Some BRT Settings. 1.3. Complementation Theorems. 1.4. Thin Set Theorems. 1.1. General Formulation. Before presenting the precise formulation of Boolean Relation Theory (BRT), we give two examples of assertions in BRT that are of special importance for the theory. DEFINITION 1.1.1. N is the set of all nonnegative integers. A\B = {x: x A x B}. THIN SET THEOREM. Let k 1 and f:N k N. There exists an infinite set A N such that f[A k ] N. COMPLEMENTATION THEOREM. Let k 1 and f:N k N. Suppose that for all x N k , f(x) > max(x). There exists an infinite set A N such that f[A k ] = N\A. These two theorems are assertions in BRT. In fact, the complementation theorem has the following sharper form. COMPLEMENTATION THEOREM (with uniqueness). Let k 1 and f:N k N. Suppose that for all x N k , f(x) > max(x). There exists a unique set A N such that f[A k ] = N\A. Furthermore, A is infinite. We will explore the Thin Set Theorem and the Complementation Theorem in sections 1.3, 1.4. At this point we analyze their logical structure. DEFINITION 1.1.2. A multivariate function on N is a function whose domain is some N k and whose range is a subset of N. A strictly dominating function on N is a multivariate function on N such that for all x N k , f(x) > max(x). We define MF as the set of all multivariate functions on N, SD as the set of all strictly dominating functions on N, and INF as the set of all infinite subsets of N.
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2 DEFINITION 1.1.3. Let f MF, where dom(f) = N k . For A N, we define fA = f[A k ]. The notation fA is very convenient. It avoids the unnecessary use of explicit mention of arity or dimension. It is used throughout this book. Using this notation, we can restate our two theorems as follows. THIN SET THEOREM. For all f MF there exists A INF such that fA N. COMPLEMENTATION THEOREM. For all f SD there exists A INF such that fA = N\A. Note that in the Thin Set Theorem, we use the family of multivariate functions MF, and the family of sets INF. In the Complementation Theorem, we use the family of multivariate functions SD, and the family of sets INF. In BRT terminology this will be expressed by saying that the Thin Set Theorem is an instance of IBRT (inequational BRT) on the BRT setting (MF,INF), and the Complementation Theorem is an instance of EBRT (equational BRT) on the BRT setting (SD,INF). Note that we can regard the condition fA N as a Boolean inequation in fA,N. We also regard the condition fA = N\A as a Boolean equation in fA,N. Here N plays the role of the universal set in Boolean algebra. From this perspective, fA N is a Boolean inequation in fA, and fA = N\A is a Boolean equation in A,fA. The fact that N should play the role of the universal set
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1.1GenForm062610 - 1 CHAPTER 1 INTRODUCTION TO BRT 1.1....

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