Logic and Proofs

🏛️ Introduction to Proofs: Building Your Logic Castle

The Story of the Logic Detective 🔍

Imagine you’re a detective. Not just any detective—a Logic Detective. Your job? To prove things are TRUE beyond any doubt. You can’t just say “I think the butler did it.” You need evidence. You need steps. You need PROOF.

That’s exactly what geometry proofs are about. They’re your detective toolkit for proving mathematical truths!

🎯 What Are Logical Statements?

A logical statement is a sentence that is either TRUE or FALSE—never both, never “maybe.”

Think of It Like a Light Switch 💡

• ON = TRUE
• OFF = FALSE
• There’s no “kinda on” or “sort of off”

Examples of Logical Statements

NOT Logical Statements

These are not logical statements because they can’t be judged true or false:

• “Wow, that’s a cool shape!” (opinion)
• “Is that a triangle?” (question)
• “Draw a circle” (command)

🎭 Counterexamples: The Proof Breaker

A counterexample is ONE example that proves a statement is FALSE.

The Detective’s Secret Weapon

Imagine someone says: “All birds can fly.”

You only need ONE bird that can’t fly to prove them wrong:

• 🐧 Penguin! Can’t fly. Statement = FALSE!

How Counterexamples Work

Statement: "All numbers ending in 2 are even"
Test it: 2, 12, 22, 32... all even!
Result: TRUE ✅

Statement: "All prime numbers are odd"
Counterexample: 2 is prime AND even!
Result: FALSE ❌

One counterexample is enough to destroy any claim!

🏗️ Postulates and Axioms: The Building Blocks

Postulates (also called axioms) are statements we accept as TRUE without proof. They’re our starting points.

Like the Rules of a Game

When you play chess, you don’t ask “Why does the knight move in an L?” You just accept it. That’s the rule.

In geometry, postulates are our rules.

Famous Geometry Postulates

Simple Example

Postulate: Through any two points,
           there is exactly one line.

Point A •————————————• Point B

You can ONLY draw ONE straight line
connecting A and B. Not two. Not zero.
Exactly ONE.

🏆 Theorems and Corollaries: Proven Truths

Theorems: The Big Discoveries

A theorem is a statement that HAS BEEN PROVEN true using:

• Postulates
• Definitions
• Other theorems

The Detective’s Case Files

Think of a theorem like a solved case. Once proven, you can use it to solve other cases!

Famous Theorem:

If two angles are vertical angles, then they are equal.

        \  /
         \/  ← These two angles
         /\    are EQUAL!
        /  \

Corollaries: The Bonus Discoveries

A corollary is a theorem that follows EASILY from another theorem. It’s like a free bonus!

Example:

• Theorem: Vertical angles are equal.
• Corollary: If two lines intersect, they form two pairs of equal angles.

The corollary is almost obvious once you know the theorem!

➡️ Direct Proof: The Straight Path

A direct proof goes from what you KNOW to what you WANT TO PROVE in a straight line.

Like Following a Trail of Breadcrumbs 🍞

START → Step 1 → Step 2 → Step 3 → GOAL!

Example: Prove that if a number is even, its square is even

Given: n is an even number

Step 1: n = 2k (definition of even)
        where k is some whole number

Step 2: n² = (2k)² = 4k²

Step 3: n² = 2(2k²)

Step 4: Since n² = 2 × (something),
        n² is even! ✅

PROVEN! 🎉

🔄 Indirect Proof: The Surprise Twist

An indirect proof (also called proof by contradiction) proves something is TRUE by showing the opposite is IMPOSSIBLE.

The Detective’s Trick

Instead of proving “The butler DID do it,” you prove “If the butler DIDN’T do it, then impossible things would happen.”

How It Works

Goal: Prove statement P is true

Step 1: Assume P is FALSE

Step 2: Follow the logic...

Step 3: Reach something IMPOSSIBLE
        (like 1 = 2, or a contradiction)

Step 4: Since assuming "P is false" leads
        to nonsense, P must be TRUE! ✅

Example: Prove √2 is not a fraction

Assume √2 IS a fraction: √2 = a/b
(where a/b is in lowest terms)

Square both sides: 2 = a²/b²

So: a² = 2b²

This means a² is even, so a is even.
Let a = 2c

Then: (2c)² = 2b²
      4c² = 2b²
      2c² = b²

So b² is even, and b is even too!

BUT WAIT! 🚨
If BOTH a and b are even, then a/b
was NOT in lowest terms!

CONTRADICTION! Our assumption was wrong.
Therefore, √2 is NOT a fraction. ✅

📝 Two-Column Proof Format: The Organized Detective

A two-column proof organizes your reasoning into two columns:

• Statements (what you claim)
• Reasons (why it’s true)

The Template

Real Example: Prove angles 1 and 3 are equal

Given: Lines AB and CD intersect at point E

     A
      \  1
       \ ∠
    2 ∠ E ∠ 4
       /∠
      / 3
     B

PROVEN! ✅

🗺️ The Proof Flow

graph TD
    A["Start with GIVEN info"] --> B["Use Definitions"]
    B --> C["Apply Postulates/Axioms"]
    C --> D["Use Known Theorems"]
    D --> E["Reach Your CONCLUSION"]
    E --> F["Q.E.D. - Proof Complete!"]

🎮 Your Proof Toolkit Summary

🚀 You’re Now a Logic Detective!

Remember:

• Every proof is like solving a mystery
• Start with what you KNOW (given information)
• Use your toolkit (definitions, postulates, theorems)
• Reach your conclusion step by step
• Always have a REASON for every statement

The best detectives don’t just guess—they PROVE! 🕵️‍♂️

“In mathematics, you don’t understand things. You just get used to them.” — John von Neumann

But with proofs, you’ll truly UNDERSTAND. That’s the power of logic! ✨