Propositional Logic Fundamentals

# CHAPTER 2

Propositional Logic Fundamentals #

1. Introduction #

2. Learning Objectives #

• Define what constitutes a valid "Proposition" in mathematics.

• Identify sentences that are NOT propositions.

• Assign Truth Values to discrete statements.

• Understand how single propositions combine to form compound logical architectures.

3. What is a Proposition? #

Valid Propositions:

• "Paris is the capital of France." *(True)*

• "$2 + 2 = 5$." *(False - It is wrong, but it is still a valid declarative proposition!)*

• "The server is currently online." *(True or False)*

NOT Propositions:

• "What time is it?" *(A question cannot be true or false).*

• "Read the documentation!" *(A command cannot be true or false).*

• "$x + 5 = 10$." *(This is an open sentence. We cannot determine if it is true or false without knowing the value of $x$.)*

• "This statement is false." *(A paradox. It cannot be assigned a truth value).*

4. Variables and Truth Values #

Let:

• $p =$ "The user is logged in."

• $q =$ "The file exists."

Every propositional variable possesses exactly one Truth Value:

• T (True) or 1

• F (False) or 0

If the user is currently logged in, then $p = T$. If the file is missing, then $q = F$.

5. Compound Propositions #

When we link multiple simple propositions together using Logical Operators (AND, OR, NOT), we create a Compound Proposition.

*Example:*

• $p$: "It is raining."

• $q$: "I will bring an umbrella."

• Compound: "It is raining, AND I will bring an umbrella."

6. The Programming Connection #

// JavaScript Boolean Variables (Propositions)
let p = true;  // Proposition p: "User is Admin"
let q = false; // Proposition q: "Account is Banned"

// A simple conditional evaluation based on Truth Values
if (p === true) {
    console.log("Welcome Admin.");
}

7. Logical State Evaluation Table #

For two propositions ($p$ and $q$), there are exactly $2^2 = 4$ possible combinations of reality:

8. Common Mistakes #

• Confusing facts with propositions: Remember, a proposition does not have to be scientifically accurate. "The moon is made of cheese" is a perfectly valid mathematical proposition because it can be definitively evaluated as False.

• Treating variables as numbers: In logic, $p$ does not equal 5. $p$ exclusively equals $T$ or $F$.

9. Exercises #

1. 1. Which of the following are valid propositions?

1. 2. If you have 3 distinct propositions ($p, q, r$), how many total possible True/False combinations exist? (Hint: Base-2 mathematics).

10. MCQs with Answers #

11. Interview Preparation #

• *Algorithmic translation:* "An API requires three separate Boolean flags to be passed. If you need to test every single possible combination of these flags to ensure the system doesn't crash, how many tests must you write?" *(Answer: $2^n$ combinations. With 3 flags, you must write exactly $2^3 = 8$ distinct unit tests to achieve 100% absolute logical coverage!)*

12. Summary #

13. Next Chapter Recommendation #