Lesson 4: Logical Reasoning and Truth Statements

👨🏽‍🏫 Introduction

Ever heard someone say, “If it rains, I’ll stay home” 🌧🏠? Or “I’ll pass if I study hard”? These are not just sentences — they’re logical statements. In mathematics, we use logic to connect ideas and test if statements are true or false using rules — just like a detective! 🕵🏽‍♂️🔍

This lesson will help you understand how logic works in math and real life using truth values (True/False), connectors like “AND”, “OR”, “NOT”, and conditionals like “If… then”.

You’ll learn to:

• Recognize and evaluate truth statements
• Use logical connectives (AND, OR, NOT, IF…THEN)
• Build and interpret truth tables
• Solve real-life logic puzzles and problems

🧠 Key Concepts

📌 1. Statements and Truth Values

A statement is a sentence that is either true or false, but not both.

Examples:

• “2 is an even number” → ✅ True
• “7 is a multiple of 3” → ❌ False

We use letters like p, q, and r to represent statements.

🔗 2. Logical Connectives

These are words we use to combine or modify statements:

• NOT p (¬p): Reverses the truth value (if p is true, ¬p is false)
• p AND q (p ∧ q): True only when both p and q are true
• p OR q (p ∨ q): True if at least one of p or q is true
• IF p THEN q (p → q): False only if p is true and q is false

📋 3. Truth Tables

A truth table shows all possible truth values of a logical expression.

✅ AND (p ∧ q)

✅ OR (p ∨ q)

✅ NOT (¬p)

✅ IF…THEN (p → q)

💡 Real-Life Logic Examples

• 📱 “If I charge my phone, it will turn on” (p → q)
• ☕ “I will drink tea or coffee” (p ∨ q)
• 💼 “I’ll go to work only if it’s Monday and not a holiday” (p ∧ ¬q)

🧮 Sample Problem Walkthroughs

1. Given p: “3 is a prime”, q: “4 is odd”, find ¬q and p ∧ q
   Solution:
   p = T, q = F
   ¬q = T, p ∧ q = T ∧ F = F
2. Evaluate p → q when p is false and q is true
   Answer: T (only false when p is true and q is false)
3. Construct a truth table for: ¬p ∨ q
   

   p	q	¬p	¬p ∨ q
   T	T	F	T
   T	F	F	F
   F	T	T	T
   F	F	T	T

✍🏽 Practice Exercises

🅰️ A. Truth Values:

1. Is the statement “0 is a positive number” true or false?
2. Let p: “2 is even”, q: “5 is odd”. What is p ∧ q?

🅱️ B. Truth Tables:

1. Complete the truth table for p ∨ ¬q

🅾️ C. Real-Life Logic:

1. Express this: “If I revise, I will pass” in logical form.
2. Is the statement “If 3 > 5, then 2 = 2” true?

✅ Answers and Explanations

• A1: ❌ False
• A2: p = T, q = T → p ∧ q = T

• B1:
  

  p	q	¬q	p ∨ ¬q
  T	T	F	T
  T	F	T	T
  F	T	F	F
  F	F	T	T

• C1: Let p = “I revise”, q = “I pass” → Statement is p → q
• C2: ✅ True (because p is false, and q is true → implication is true)

📌 Recap

• ✅ Logical reasoning uses connectors like AND, OR, NOT, and IF…THEN
• ✅ Truth tables help us see when statements are true or false
• ✅ Statements are either true or false — no in-between
• ✅ Use logic to analyze, prove, or predict outcomes

💭 Reflection Prompt

Have you ever made a promise or condition like “If you wash the dishes, I’ll let you play”? How can understanding truth logic help you make clearer decisions in real life? 🤔