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Full Description
Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra.
The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies.
Features
Designed to be teachable across a single semester
Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses
Offers a balanced variety of easy, moderate, and difficult exercises
Contents
1. Logic. 1.1 Introduction. 1.2. Statements and Logical Connectives. 1.3 Logical Equivalence. 1.4. Predicates and Quantifiers. 1.5. Negation. 2. Proof Techniques. 2.1. Introduction. 2.2. The Axiomatic and Rigorous Nature of Mathematics. 2.3. Foundations. 2.4. Direct Proof. 2.5. Proof by Contrapositive. 2.5. Proof by Cases. 2.6. Proof by Contradiction. 3. Sets. 3.1. The Concept of a Set. 3.2. Subset of Set Equality. 3.3. Operations on Sets. 3.4. Indexed Sets. 3.5. Russel's Paradox. 4. Proof by Mathematical Induction. 4.1. Introduction. 4.2. The Principle of Mathematical Induction. 4.3. Proof by strong Induction. 5. Relations. 5.1. Introduction. 5.2. Properties of Relations. 5.3. Equivalence Relations. 6. Introduction. 6.1. Definition of a Function. 6.2. One-To-One and Onto Functions. 6.3. Composition of Functions. 6.4. Inverse of a Function. 7. Cardinality of Sets. 7.1. Introduction. 7.2. Sets with the same Cardinality. 7.3. Finite and Infinite Sets. 7.4. Countably Infinite Sets. 7.5. Uncountable Sets. 7.6 Comparing Cardinalities.
