* The Proof Page *

Created by

* D. A. Kouba *

Department of Mathematics

University of California, Davis

With generous support from a UIIP (Undergraduate Instructional Improvement Program) grant from TRC

Proving that a mathematical statement is true can be a challenging task. It requires a careful reading of the statement, a clear understanding of the statement, a choice of method to prove the statement, and a sensible, step-by-step proof of the statement. This website is designed to assist those who are just beginning to learn both the art and science of proof. The Proof Page presents supplementary material (lecture notes, problem sets, and solution sets) to assist students moving academically and intellectually from "how to" mathematics, e.g., algebra, trigonometry, analytic geometry, differential calculus, integral calculus, vector calculus, linear algebra, and differential equations, to "why" mathematics, where the focus will be on proving both familiar and unfamiliar statements.

Webster's Collegiate Dictionary defines the word

"the process or an instance of establishing the validity of a statement especially by derivation

from other statements in accordance with accepted or stipulated principles of reasoning."

This definition implicitly suggests that a wide variety of methods and styes can be used when proving mathematical statements. In addition, learning to write steps to a proof and knowing when a proof is "complete" depend on the standards and expectations of the professor and the methods and level of difficulty of the associated textbook. Because each professor, who teaches an introductory proof course, has a unique and personalized proving style, it is in the best interest of all students using this website as a supplementary resource to always defer to their class professor when questions or differences in proof styles arise.

The methods and styles used on The Proof Page closely follow those presented in Chapters 1, 2, 3, 4, 5, and 7 in the fifth edition of A Transition to Advanced Mathematics by Douglas Smith, Maurice Eggen, and Richard St. Andre. This is a textbook used for Math 108, Introduction to Abstract Mathematics, in the Department of Mathematics at the University of California, Davis.

Here is a Math 108 website featuring, practice problems, problem sets, and solution sets.

- Section 1.1 : Propositions and Connectives; Truth Tables
- Section 1.2 : Conditionals and Biconditionals; Mathematically Equivalent Statements
- Section 1.3 : Universal Quantifier (For all x ...); Existential Quantifier (There exists x ...); Unique Existential Quantifier (There exists a unique x ...)
- Section 1.4 : Basic Proof Methods I- Direct Proof, Proof by Cases, Proof by Working Backward
- Section 1.5 : Basic Proof Methods II- Proof by Contrapositive; Proof by Contradiction; Proof of a Biconditional Statement
- Section 1.6 : Proofs Involving Quantifiers; Counterexamples
- Section 1.7 : Additional Examples of Proofs
- Section 2.1 : Basic Set Theory
- Section 2.2 : Set Operations
- Section 2.3 : Extended Set Operations; Indexed Families of Sets
- Section 2.4 : Inductive Sets; the Principle of Mathematical Induction
- Section 2.5 : Principle of Complete Induction; Well-Ordering Principle
- Section 3.1 : Cartesian Products; Relations
- Section 3.2 : Equivalence Relations
- Section 3.3 : Partitions
- Section 4.1 : Functions
- Section 4.2 : Construction of New Functions
- Section 4.3 : Functions that are Onto; Functions that are One-to-One
- Section 5.1 : Equivalent Sets; Finite Sets
- Section 5.2 : Infinite Sets- Countable (Denumerable); Uncountable
- Section 5.3 : Countable Sets
- Section 7.1 : Ordered Field Properties of the Real Numbers
- Section 7.2 : Open Sets, Closed Sets, Compact Sets, and The Heine-Borel Theorem