Clearly written and easy to understand, The Elements of Advanced Mathematics covers logic, set theory, methods of proof, and axiomatic structures, providing an excellent grounding in analytical thinking. It facilitates the transition from elementary mathematics, generally characterized by problem-solving techniques, to advanced mathematics, characterized by theory, rigor, and proofs. This text clearly identifies and explains the components and methods of advanced mathematics. Each chapter contains exercises designed to assist the reader in understanding the material.

Contenuti:

Basic Logic: Principles of Logic. Truth. "And" and "Or". "Not". "If-Then". Contrapositive, Converse, and "Iff". Quantifiers. Truth and Provability. Exercises.
Methods of Proof: What is a Proof? Direct Proof. Proof by Contradiction. Proof by Induction. Exercises.
Set Theory: Undefinable Terms. Elements of Set Theory. Venn Diagrams. Further Ideas in Elementary Set Theory. Indexing and Extended Set Operations. Exercises.
Relations and Functions: Relations . Order Relations. Functions. Combining Functions. Cantor's Notion of Cardinality. Exercises.
Axioms of Set Theory, Paradoxes, and Rigor: Axioms of Set Theory. The Axiom of Choice. Set Theory and Arithmetic. Exercises.
Number Systems: Preliminary Remarks. The Natural Number System. The Integers. The Rational Numbers. The Real Number System. The Complex Numbers. The Quaternions, the Cayley Numbers, and Beyond. Exercises.
Examples of Axiomatic Theories: Group Theory. Euclidean and Non-Euclidean Geometry. Exercises. Bibliography. Index.

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