The axioms of "equality" a = a Reflexive or Identity. The intersection of sets A and B is the set A\B = fx : x 2A^x 2Bg. function from the set of real numbers into X or there is a one-to-one function from X into the set of rational numbers. These are the "rules" that govern the use of the = sign. 3 The algebra of classes 4 Ordered pairs Cartesian products 5 Graphs 6 Generalized union and intersection 7 Sets Chapter 2 Functions 1 Introduction 2 Fundamental concepts and definitions 3 Properties of composite functions and inverse functions 4 Direct images and inverse images under functions 5 Product of a family of classes 6 The axiom of replacement Chapter 3 Relations 1 … Set Operations and the Laws of Set Theory The union of sets A and B is the set A[B = fx : x 2A_x 2Bg. Our approach is based on a widely used strategy of mathematicians: we work with speciﬁc examples and look for general patterns. Basic Laws of Set Theory. 8 CHAPTER 0. If a = b, then b = a. Symmetry. They won’t appear on an assignment, however, because they are quite dif-7. The commutative rules of addition and multiplication This section discusses operations on sets and the laws governing these set opera-tions. Statement (2) is true; it is called the Schroder-Bernstein Theorem. The Formal Rules of Algebra Summary of the formal rules of algebra on the set of real numbers 1. 2. a = c. Transitivity . a = b. and . b = c, then . Linear algebra is one of the most applicable areas of mathematics. This study leads to the deﬁnition of modiﬁed addition and multiplication operations on certain ﬁnite subsets of the integers. Alternate notation: A B. In the next two chapters we will see that probability and statistics are based on counting the elements in sets and manipulating set operations. This book is directed more at the former audience Laws of Algebra 1) Idempotent Laws: Let A be a set A A = A A A = A 2) Identity Laws: Let A be a set and U be a Universal set A = A A U = A 3) Commutative Laws: Let A and B be sets A B = B A A B = B A 4) Associative Laws: Let A , B and C be sets A (B C) = (A B) C If . troduction to abstract linear algebra for undergraduates, possibly even ﬁrst year students, specializing in mathematics. It is quite clear that most of these laws resemble or, in fact, are analogues of laws in basic algebra and the algebra of propositions. Subsection 4.2.2 Proof Using Previously Proven Theorems. The set di erence of A and B is the set AnB = fx : x 2A^x 62Bg. Once a few basic laws or theorems have been established, we frequently use them to prove additional theorems. INTRODUCTION ﬁcult to prove. It is used by the pure mathematician and by the mathematically trained scien-tists of all disciplines. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. The symmetric di erence of A and B is A B = (AnB)[(B nA). These are fundamental notions that will be used throughout the remainder of this text.

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