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

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