## Propositional Logic (Truth Tables and Their Usage)

By a mathematical statement (or statement, or proposition) we mean  a declarative sentence that can be classified as either true or false, but not both. For example, the sentences $$1+3=4, \quad 1+3=5, \quad \text{July is not a month}$$ can be accepted

## Quantifiers and Predicate Logic

Variables in mathematical statements can be quantified in different ways. First, the symbol $\forall$ is called a universal quantifier and is used to express that a variable may take on any value in a given collection. For example, $\forall x$  is a

## Mathematical Proofs (Using Various Methods)

Are you someone who relies on logic and evidence for solving problems? Mathematical proofs will help you refine and take advantage of this valuable way of thinking as it applies to mathematics, and potentially other areas such as philosophy and

## Logical Discourse Using Rules of Inference

Logical Discourse In this section we discuss axiomatic systems and inference rules for quantified statements. To give example, we carry out a simple logical discourse for incidence geometry involving points, lines, and incidence.    Axiomatic Systems An axiomatic (or formal) system

## Set Theory (Basic Theorems with Many Examples)

We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and Cartesian products. We also demonstrate how to work with families of sets. For a brief discussion of the reviews of (elementary) Halmos’ Naive Set

## Binary Relations (Types and Properties)

Let $X$ be a set and let $X\times X=\{(a,b): a,b \in X\}.$ A (binary) relation $R$ is a subset of $X\times X$. If $(a,b)\in R$, then we say $a$ is related to $b$ by $R$. It is possible to have both $(a,b)\in R$ and

## Equivalence Relations (Properties and Closures)

Equivalence Relations We discuss the reflexive, symmetric, and transitive properties and their closures. The relationship between a partition of a set and an equivalence relation on a set is detailed. We then give the two most important examples of equivalence

## Partial Order Relations (Mappings on Ordered Sets)

We discuss many properties of ordered sets including Noetherian ordered sets and order ideals. We also detail monotone mappings and isomorphisms between ordered sets. Ordered Sets Throughout we assume $(X,\geq)$ is an ordered set.  By this we mean that $X$

## Well-Founded Relations (and Well-Founded Induction)

Well-Founded induction is a generalization of mathematical induction. Well-Founded Induction Definition. Let $\longrightarrow$ be a relation on $X.$ 1) If $A\subseteq X$ and $a\in A,$ then $a$ is called a $\longrightarrow$-minimal element of $A$ if  there does not exist \$b\in

## Confluent Relations (using Reduction Relations)

We discuss confluent relations; in particular, we prove Newman’s Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation. We also give a generalization of Newman’s lemma based on