Welcome to Week 1!
Hello everyone! I'm excited to start this math journey with you. This week, we laid the groundwork for the course and began exploring the fascinating world of logic. Below you'll find the syllabus and resources to help you succeed in Chapter 1, sections 1 and 2. Let's make this a great semester!
Course Syllabus
Please take some time to review the course syllabus. It contains important information about grading, assignments, and expectations. You can find the PowerPoint presentation from our first class below, which covers key details from the syllabus. Understanding these details is crucial for your success in this course.
Chapter 1: Propositional Logic
This week, we began our exploration of Propositional Logic. This involves understanding statements that can be either true or false. These statements, called propositions, are the building blocks of more complex logical arguments.
Key concepts we covered include:
- Propositions: Statements that are either true or false, but not both. Examples: "The sky is blue" or "2 + 2 = 4".
- Logical Operators: Symbols used to combine propositions. We specifically examined:
- Negation (¬): Reverses the truth value of a proposition. If $p$ is true, then $¬p$ is false, and vice-versa.
- Conjunction (∧): Represents "and". $p ∧ q$ is true only if both $p$ and $q$ are true.
- Disjunction (∨): Represents "or". $p ∨ q$ is true if either $p$ or $q$ (or both) are true.
- Exclusive Or (⊕): Represents "either... or... but not both". $p ⊕ q$ is true if either $p$ is true or $q$ is true, but not if they are both true.
- Conditional (→): Represents "if... then...". $p → q$ is only false when $p$ is true and $q$ is false.
- Biconditional (↔): Represents "if and only if". $p ↔ q$ is true when $p$ and $q$ have the same truth value (both true or both false).
Truth Tables
We also learned how to construct truth tables, which are essential for analyzing the truth values of compound propositions. A truth table lists all possible combinations of truth values for the individual propositions and the resulting truth value of the entire expression.
For example, the truth table for $p ∧ q$ (p and q) is:
| p | q | p ∧ q |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
Excel and Truth Tables
To further enhance your understanding, I've provided a resource that demonstrates how to create truth tables using Excel. This is a valuable skill for visualizing logical operations and checking your work. I encourage you to try it out yourself! Pay close attention to the comments within the Excel sheet, as there's a minor correction needed for one of the formulas. Spotting and correcting errors is a great way to learn!
Remember: Practice is key! The more you work with propositional logic and truth tables, the more comfortable you'll become. Don't hesitate to ask questions in class or during office hours.
Downloads
Let's have a fantastic week 2!