Course Major Units

Specific Student Learning Outcomes for Units

The student will be able to…

Note:  Methods of proofs and applications of proofs are emphasized throughout the course.

Logic - Propositional Calculus

Doer

3.1 Propositions and Logical Operators

3.2 Truth Tables and Propositions Generated by a Set

3.3 Equivalence and Implication

Kwong

2.1 Propositions

2.2 Conjunctions and Disjunctions

2.3 Implications

2.4 Biconditional

2.5 Logical Equivalences

Levin

3.1 Propositional Logic

      Truth Tables

      Logical Equivalence

      Deductions

Lipschutz

4.1 Introduction

4.2 Propositions and Compound

      Statements

4.3 Basic Logical Operations

4.4 Propositions and Truth Tables

4.5 Tautologies and Contradictions

4.6 Logical Equivalence

4.7 Algebra of Propositions

4.8 Conditional and Biconditional Statements

4.9 Arguments

● Use statements, variables, and logical connectives to translate between English and formal logic.

● Use a truth table to prove the logical equivalence of statements.

● Identify conditional statements and their variations.

● Identify common argument forms.

● Use truth tables to prove the validity of arguments.

Logic - Predicate Calculus

Doer

3.8 Quantifiers

Kwong

2.6 Logical Quantifiers

Levin

3.1 Beyond Propositions

Logic

Lipschutz

4.10 Propositional Functions, Quantifiers

4.11 Negation of Quantified Statements

1.3 Venn Diagrams

● Use predicates and quantifiers to translate between English and formal logic.

● Use Euler diagrams to prove the validity of arguments with quantifiers.

Logic – Proofs

Doer

3.5 Mathematical Systems and Proofs

3.6 Propositions over a Universe

3.7 Mathematical Induction

3.9 A Review of Methods of Proof

Kwong

1.4 Proving Identities

3.1 An Introduction to Proof Techniques 

3.2 Direct Proofs

3.3 Indirect Proofs

3.4 Mathematical Induction: An Introduction 

3.5 More on Mathematical Induction 3.6 Mathematical Induction: The Strong Form

Levin

  2.5 Induction

       Formalizing Proofs

 
  3.2 Proofs 

Lipschutz

1.8 Mathematical Induction

Pick Sections from chapter 11

Properties of the Integers

● Construct proofs of mathematical statements - including number theoretic statements - using counter-examples, direct arguments, division into cases, and indirect arguments.

● Use mathematical induction to prove propositions over the positive integers.

Set Theory

Doer

1.1 Set Notation and Relations

1.2 Basic Set Operations

1.3 Cartesian Products and Power Sets

4.1 Methods of Proof for Sets

4.2 Laws of Set Theory

Kwong

4.1 An Introduction

4.2 Subsets and Power

4.3 Unions and Intersections

4.4 Cartesian Products

Levin

0.3 Sets

      Notation

      Relationships Between Sets 

      Operations On Sets

      Venn Diagrams

Lipschutz

1.1 Introduction

1.2 Sets and Elements, Subsets

1.3 Venn Diagrams

1.4 Set Operations

1.7 Classes of Sets, Power Sets, Partitions

● Exhibit proper use of set notation, abbreviations for common sets, Cartesian products, and ordered n-tuples. ● Combine sets using set operations.  

● List the elements of a power set.  

● Lists the elements of a cross product.

● Draw Venn diagrams that represent set operations and set relations.

● Apply concepts of sets or Venn Diagrams to prove the equality or inequality of infinite or finite sets.

● Create bijective mappings to prove that two sets do or do not have the same cardinality

Functions and Relations

Doer

6.1 Basic Definitions

6.2 Graphs of Relations on a   

      Set

6.3 Properties of Relations

7.1 Definition and Notation

7.2 Properties of Functions

7.3 Function Composition

Kwong

6.1 Functions: An Introduction

6.2 Definition of Functions

6.3 One-to-One Functions

6.4 Onto Functions

6.5 Properties of Functions 

6.6 Inverse Functions

6.7 Composite Functions

7.1 Definition of Relations

7.2 Properties of Relations

7.3 Equivalence Relations

7.4 Partial and Total Ordering

Levin

0.4 Functions

      Describing Functions
      Surjections, Injections, 
      and Bijections

      Image and Inverse Image

 
Lipschutz

3.1 Introduction

3.2 Functions

3.3 One-to-One, Onto, and Invertible Functions

2.1 Introduction

2.2 Product Sets

2.3 Relations

2.4 Pictorial Representations of Relations

2.6 Types of Relations

2.8 Equivalence Relations

2.9 Partial Ordering Relations

14.3 Hasse Diagrams of Partially Ordered Sets

● Identify a function's rule, domain, codomain, and range.

● Draw and interpret arrow diagrams. 

● Prove that a function is well-defined, one-to-one, or onto.

● Given a binary relation on a set, determine if two elements of the set are related.

● Prove that a relation is an equivalence relation and determine its equivalence classes.

● Determine if a relation is a partial ordering.

Counting Theory

Doer

2.1 Basic Counting Techniques - The Rule of Products

2.2 Permutations

2.4 Combinations and the Binomial Theorem

Kwong

8.1 What is Combinatorics?

8.2 Addition and Multiplication Principles

8.3 Permutations

8.4 Combinations

8.5 The Binomial Theorem

Levin

1.1 Additive and Multiplicative  

      Principles

      Counting with Sets
      Principle of  Inclusion/Exclusion

1.2 Binomial Coefficients

      Pascal’s Triangle

1.3 Combinations and   

      Permutations

1.4 Combinatorial Proofs

     Patterns in Pascal’s   

     Triangle

      More Proofs

Lipschutz

1.6 Finite Sets, Counting Principle

3.7 Cardinality

● Use the multiplication rule, permutations, combinations, and the pigeonhole principle to count the number of elements in a set.

● Apply the Binomial Theorem to counting problems.

Graph Theory

Doer

9.1 Graphs - General Introduction

9.4 Traversals: Eulerian and Hamiltonian Graphs

10.1 What Is a Tree?

10.2 Spanning Trees

Kwong

No sections

Levin

4 Graph Theory

   4.1 Basics

   4.4 Euler Paths and Circuits  

        Hamilton Paths

Lipschutz

8.2 Graphs and Multigraphs

8.3 Subgraphs, Isomorphic and Homeomorphic Graphs

8.4 Paths, Connectivity

8.5 Traversable and Eulerian Graphs, Bridges of Königsberg

8.8 Tree Graphs

● Identify the features of a graph using definitions and proper graph terminology.

● Prove statements using the Handshake Theorem.

● Prove that a graph has an Euler circuit.

● Identify a minimum spanning tree.

Boolean Algebra

Doer

13.1 Posets Revisited

13.2 Lattices

13.3 Boolean Algebras

Kwong

No sections

Levin

No sections

Lipschutz

15.1 Introduction

15.2 Basic Definitions

15.5 Boolean Algebras as Lattices

14.2 Ordered Sets

14.8 Lattices

14.9 Bounded Lattices

14.10 Distributive Lattices

14.11 Complements, Complemented Lattices

● Define Boolean Algebra.

● Apply its concepts to other areas of discrete math.

● Apply partial orderings to Boolean algebra.

Recurrence Relations

Doer

8.2 Sequences

8.3 Recurrence Relations

Kwong

No sections

Levin

2 Sequences

   2.1 Describing Sequences

   2.2 Arithmetic and Geometric 

         Sequences

        Sums of Arithmetic and Geometric Sequences

   2.4 Solving Recurrence Relations

         The Characteristic Root

         Technique

Lipschutz

3.6 Recursively Defined Functions

6.6 Recurrence Relations

6.7 Linear Recurrence Relations with Constant Coefficients

6.8 Solving Second-Order Homogeneous Recurrence Relations

● Give explicit and recursive descriptions of sequences. ● Solve recurrence relations.