Saturday, February 29, 2020
Friday, February 28, 2020
Discrete Mathematics - An Introduction to Proofs with Set Theory
The notes for this section are available here.
Discrete Mathematics - General Recursive Definitions and Structural Induction
The notes for this section are available here.
Discrete Mathematics - Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients
The notes for this section are available here.
Thursday, February 27, 2020
Wednesday, February 26, 2020
Discrete Mathematics - Solving Recurrence Relations by Iteration
The notes for this section are available here.
Discrete Mathematics - Defining Sequences Recursively
The notes for this section are available here.
Discrete Mathematics - Strong Mathematical Induction and the Well-Ordering Principle for the Integers
The notes for this section are available here.
Discrete Mathematics - Part 2: Mathematical Induction
The notes for this section are available here.
Discrete Mathematics - Part 1: Mathematical Induction
The notes for this section are available here.
Discrete Mathematics - Algorithms
Algorithms in general, then the Division and Euclidean Algorithms are discussed in this section. The notes for this section are available here.
Discrete Mathematics - The Square Root of 2 is Irrational and There are Infinitely Many Prime Numbers
In this section, we will show that the square root of 2 is irrational and that there are infinitely many prime numbers. We will also continue to discuss indirect methods to proving statements. The notes for this section are available here.
Discrete Mathematics - Contradiction and Contraposition
In this section, proofs by contradiction and contraposition will be discussed. The notes for this section are available here.
Tuesday, February 25, 2020
Monday, February 24, 2020
Discrete Mathematics - Part 5: Direct Proof and Counterexample (Floor and Ceiling)
The notes for this section are available here.
Discrete Mathematics - Part 4: Direct Proof and Counterexample (Division into Cases, Quotient Remainder Theorem)
The notes for this section are available here.
Discrete Mathematics - Part 3: Direct Proof and Counterexample (Divisibility)
In this section, we will continue to do more direct proof and counterexamples involving divisibility.
Let a and b be integers. We say that a divides b, read a | b, if b = ak for some integer k.
The notes for this section are available here.
Let a and b be integers. We say that a divides b, read a | b, if b = ak for some integer k.
The notes for this section are available here.
Discrete Mathematics - Part 2: Direct Proof and Counterexample (Rational Numbers)
In this section, we will do more direct proofs and counterexamples with an emphasis on learning about rational numbers. The notes for this section are available here.
Discrete Mathematics - Part 1: Direct Proof and Counterexample
In this section, we learn to write proofs for existential statements and universal statements. We also learn how to write counterexamples.
It is very important to remember that the negation of a universal statement is an existential statement. This means that to disprove a universal statement, we only need to find ONE counterexample!
Example:
Proposition: For every positive integer n, n^2 is odd.
Counterexample: Note that 2 is a positive integer, but 2^2 = 4 is even. Therefore, the proposition is false.
You DO NOT want to use a generic variable to show that this proposition is false, because it logically opposes the negation of a universal statement being an existential one!
The notes for this section are available here.
It is very important to remember that the negation of a universal statement is an existential statement. This means that to disprove a universal statement, we only need to find ONE counterexample!
Example:
Proposition: For every positive integer n, n^2 is odd.
Counterexample: Note that 2 is a positive integer, but 2^2 = 4 is even. Therefore, the proposition is false.
You DO NOT want to use a generic variable to show that this proposition is false, because it logically opposes the negation of a universal statement being an existential one!
The notes for this section are available here.
Discrete Mathematics - Arguments with Quantified Statements
Universal modus ponens and universal modus tollens are discussed in this section. The validity of arguments is studied extensively in this section as well. The notes for this section are available here.
Discrete Mathematics - Statements with Multiple Quantifiers
The notes for this section are available here.
Discrete Mathematics - Part 2: Predicates and Quantified Statements
The notes for this section are available here.
Discrete Mathematics - Part I: Predicates and Quantified Statements
The notes for this section are available here.
Discrete Mathematics - Number Systems and Circuits for Addition
In this section, we discuss binary representation and arithmetic. Two's complements, representation of negative integers in a computer along with adding them, 8-bit representations, and hexadecimal notation are studied as well. The notes for this section are available here.
Discrete Mathematics - Digital Logic Circuits
In this section, we examine the application of logic to digital circuits. The notes for this section are available here.
Discrete Mathematics - Valid and Invalid Arguments
In this section, valid and invalid argument forms are discussed. The notes for this section can be found here.
**Note: There is an error for elimination in the document. It should be therefore q instead of therefore not q under p v q, not p. Also, it should be therefore p instead of therefore not p under p v q, not q. My apologies for this error.
Elimination
p v q p v q
~p ~q
therefore q therefore p
Discrete Mathematics - Conditional Statements
In this section, we learn about conditional statements along with the converse, inverse, contrapositive, and biconditional. We learn about their truth tables as well as notation and identities. In addition, we will learn that p implies q is logically equivalent to not p or q which is one of the most important identities involving statements. Necessary and sufficient conditions are studied in this section as well. The notes for this section can be found here.
Discrete Mathematics - Logical Form and Logical Equivalence
In this section of Discrete Mathematics, one will learn about statements, truth tables, logical equivalence, DeMorgan's laws, negation, conjunction (and) and disjunction (or) statements. The notes for this section can be found here.
Discrete Mathematics - The Language of Relations and Functions
In this section of Discrete Mathematics, one learns about relations and functions. We will see that relations are in fact the backbone of functions. Also, every function is a relation, but not every relation is necessarily a function. The notes for this section can be found here.
Discrete Mathematics - The Language of Sets
In this section of Discrete Mathematics, one learns introductory set theory. Further concepts in set theory will be discussed later. Set-roster notation, set-builder notation, subsets, and Cartesian products are discussed. The notes for this section can be found here.
Discrete Mathematics - Variables
In the Variables section of Discrete Mathematics, one learns how to use variables when communicating mathematics. In addition, universal statements, existential statements, and conditional statements are studied. The notes for this section can be found here.
Here are some additional exercises to try:
1) Use variables to rewrite the following sentence more formally
Any real number raised to the fourth power is nonnegative.
2) Use variable to rewrite the following sentence more formally
Do there exist numbers with the property that the difference of their cubes equals the cube of their difference?
Here are some additional exercises to try:
1) Use variables to rewrite the following sentence more formally
Any real number raised to the fourth power is nonnegative.
2) Use variable to rewrite the following sentence more formally
Do there exist numbers with the property that the difference of their cubes equals the cube of their difference?
Expressions Containing Sums or Differences of Cubes - Supplementary Notes
Here are the supplementary notes for this section. Enjoy!
Sunday, February 23, 2020
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