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Unformatted text preview: Mathematical Structures Horst R. Thieme Arizona State University, Course Notes Spring 2010. c updated March 5, 2010 2 Mathematics: what is that? In ancient Greek, (math emata) means science, knowledge. The words (mathematik e techn e) mean something like scientific craft. According to Access Science , the word mathematics stems from a root that means learnable knowledge and means the deductive study of shape, quantity and dependence. Other dictionaries and encyclopedia give different explanations. For me, mathematics is the symbolic representation and analysis of reality. Let me back this up. Somewhere out there is a universal reality (Platos realm of ideas, Kants realm of noumena, reality as seen by God). Then there is the socalled real world, which is an unreflected model of the universal reality created by our senses. In order to understand and cope with reality better, we also create re flected models of different kinds: Physical and electric models like miniature planes, air tunnels, circuits etc. Graphical models like maps, structural designs, schematic representa tion etc. Conceptual models mainly consisting of verbal descriptions Symbolic models like in Mathematics (or Chemistry). Another form of symbolic modeling is the representation of music by notes. There also are the metamodels of philosophy and theology. i ii The material of Mathematics are concepts like numbers, sets, relations etc. In dealing with these concepts, a mathematician uses formal tools like proofs, logic, symbols, and deductive reasoning. Deductive reasoning consists of topdown thinking as opposed to the bottomup thinking of inductive reasoning in experimental sciences. The mathematician also uses informal tools: intuition, heuristics, fantasy, patience, endurance, resilience to frustration. Heuristics: guide to finding scientific knowledge. Contents 1 Logic and Proof 1 1.1 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Elementary logical equivalences . . . . . . . . . . . . . 5 1.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Several quantifiers . . . . . . . . . . . . . . . . . . . . 16 1.2.2 Negation of statements with quantifiers . . . . . . . . . 18 1.2.3 Quantifiers and connectives . . . . . . . . . . . . . . . 20 1.3 Techniques of Proof: I. Direct proof, deduction, contraposition 24 1.4 Techniques of Proof: II. Contradiction, Case analysis . . . . . 28 1.4.1 Case analysis . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.2 How to prove a statement of the form p ( q r ) . . . 32 2 Sets and Functions 35 2.1 Basic Set Operations . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3 Ordered pairs and Cartesian products . . . . . . . . . . . . . . 51 2.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582....
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This note was uploaded on 03/09/2010 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Math

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