Proofs 101

Author Joseph Kirtland received his BS from Syracuse University and PhD in mathematics from the University of New Hampshire. He joined the faculty of Marist College in 1992. A highly respected teacher, he has been selected ten times by the students for the Faculty Recognition Award in the School of Computer Science and Mathematics. In 2000, he was presented with the Marist College Board of Trustee's Distinguished Teaching Award, and in 2002, he received the Award for Distinguished College or University Teaching of Mathematics given by the Metropolitan New York section of the Mathematical Association of America. In addition, his first book Identification Numbers and Check Digit Schemes won the Mathematical Association of America’s Beckenbach Book Prize in 2002. Dr. Kirtland’s professional interests include finite and infinite group theory, applications of group theory, and mathematics education. He also enjoys poetry (Philip Larkin and Mary Oliver are two of his favorite poets), hiking (he and his wife have hiked nearly ever trail in the Catskill Mountains), and cycling (his bike can often be found in northern Dutchess County, NY).

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.