Category: Resources(Page 1 of 2)

Theorem NT 5.1: Every natural number $n>1$ is either prime or divisible by a prime.

Theorem NT 5.2: Suppose $p$ is prime and $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are $n$ integers, where $n \geq 2$. If $p \mid a_{1} \cdot a_{2} \cdot a_{3} \cdot \ldots \cdot a_{n},$ then $p \mid a_{i}$ for at least one of the $a_{i}(1 \leq i \leq n)$.

Theorem NT 5.3: If $n$ is an integer greater than 1 then $n$ can be written as a product of primes.
(HINT: Prove using strong induction. Consider two cases, when $k+1$ is prime, and when it is composite)

Hi everyone,

On Tuesday we introduced the idea of greatest common divisor and we looked at several theorems about properties of the gcd.

Definition. The greatest common divisor of integers $a$ and $b$, denoted $\gcd(a,b)$, is the largest integer that divides both $a$ and $b$.

In your homework, you are asked to prove propositions that involve the gcd. It may help to keep the following in mind:

To prove that a number $x$ is the gcd of $a$ and $b$

We need to show two things:

1.  $x$ is a common divisor of $a$ and $b$ (that is, $x|a$ and $x|b$)

2.  if  $y|a$ and $y|b$, then $x\geq y$ (âif $y$ is a common divisor of $a$ and $b$, then $x\geq y$â

Here is an example, so we can see how it works in practice:

Proposition. If $a, b$ are integers then $\gcd(a,b) = \gcd(a+b,b)$.

VIDEO: Example – proof with gcd

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 8: Negating Statements, Counting Lists

Topic. This lesson covers:

1. Sec 2.10 Negating Statements
2. Sec 3.1 Counting Lists
3. Sec 3.2 Factorials

Learning Outcomes.

• Apply rules for negating various types of statements, both in formal logic and natural language
• Count collections of lists with various properties

Homework. There is one WeBWorK assignment on today’s material (Note: some of the material in this assignment will be covered in the next lesson):

1. WeBWorK: WeBWorK Assignment5-Sec3.1-3.4

Lecture Notes:

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 6: Biconditionals, Truth Tables, and Logical Equivalence

Topic. This lesson covers:

1. Sec 2.4 Biconditional Statements
2. Sec 2.5 Truth tables for Statements
3. Sec 2.6 Logical equivalence

Learning Outcomes.

• Identify instances of biconditional statements in both natural language and first-order logic, and translate between them.
• Construct truth tables for statements.
• Determine logical equivalence of statements using truth tables and logical rules.

Homework. There is one WeBWorK assignment on today’s material:

1. WeBWorK: Assignment3-Sec2.1-2.6

Lecture Notes:

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 4: Indexed Sets

Topic. This lesson covers:

• Sec 1.8: Indexed Sets

Learning Outcomes.

• Take unions and intersections of collections of sets indexed by the natural numbers or other sets.

Homework. There is 1 written assignment on today’s material:

1. Homework: Section 1.8 p.29: 3, 5, 6, 8

Lecture Notes:

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 2: Cartesian Products and Subsets

Topic. This lesson covers:

• Sec 1.2: Cartesian Products
• Sec 1.3: Subsets

Learning Outcomes.

• Identify and manipulate ordered pairs and Cartesian products of sets.
• Identify and manipulate subsets of sets.

WeBWorK. There is 1 WeBWorK assignment on today’s material:

1. Assignment1-Sec1.2-1.3

Lecture Notes:

Your first WeBWorK assignment is due on Tuesday, September 8th, at the start of class, and will cover the material from Lesson 2.  Hereâs what you have to do:

Assignment.  To get started , you must complete the following three steps.