Advanced Class Notes: DeMorgan’s Laws (Java Boolean Logic)

🎯 Learning Objectives

Understand DeMorgan’s Laws mathematically and logically
Apply DeMorgan’s Laws to simplify complex Boolean expressions
Convert negated compound conditions into equivalent forms
Use DeMorgan’s Laws in real Java programs (if statements, loops, validation)
Recognize common AP CSA test scenarios involving logical negation

🧠 Review: Boolean Logic Basics

In Java:

Operator	Meaning
`&&`	AND
`\|\|`	OR
`!`	NOT (negation)

Boolean expressions evaluate to:

true
false

⚡ What are DeMorgan’s Laws?

DeMorgan’s Laws allow us to move a NOT operator across logical expressions.

NOT (AND) becomes OR
NOT (OR) becomes AND

Key idea: When a NOT goes inside parentheses, it flips && ↔ || and negates each piece.

⭐ DeMorgan Law #1

!(A && B) == (!A || !B)

Meaning: NOT (A AND B) equals (NOT A OR NOT B)

Java Example

boolean result = !(x > 10 && y < 5);

// Equivalent:
boolean result2 = (x <= 10 || y >= 5);

⭐ DeMorgan Law #2

!(A || B) == (!A && !B)

Meaning: NOT (A OR B) equals (NOT A AND NOT B)

Java Example

boolean result = !(age < 18 || score < 70);

// Equivalent:
boolean result2 = (age >= 18 && score >= 70);

📊 Truth Table Verification

Example: !(A && B)

A	B	A && B	!(A && B)	!A \|\| !B
T	T	T	F	F
T	F	F	T	T
F	T	F	T	T
F	F	F	T	T

Notice: !(A && B) and !A || !B always match.

🚀 Advanced Usage in Java Programming

1️⃣ Simplifying Complex Conditions

Students often write:

if(!(score >= 70 && attendance >= 90))
{
    System.out.println("Not eligible");
}

Rewrite using DeMorgan:

if(score < 70 || attendance < 90)
{
    System.out.println("Not eligible");
}

✅ Easier to read ✅ Easier to debug ✅ Less nesting

2️⃣ Validation Logic (Common AP CSA Pattern)

Original:

if(!(username.equals("admin") || password.equals("1234")))
{
    // deny access
}

DeMorgan form:

if(!username.equals("admin") && !password.equals("1234"))
{
    // deny access
}

3️⃣ Loop Control Conditions

Original:

while(!(input.equals("quit") || attempts > 3))
{
    // keep looping
}

Rewrite:

while(!input.equals("quit") && attempts <= 3)
{
    // keep looping
}

4️⃣ Nested Logic Simplification

Original:

if(!(x > 0 && (y > 10 || z == 5)))
{
    // ...
}

Step-by-step:

// Step 1: !(A && B) -> !A || !B
// (x <= 0) || !(y > 10 || z == 5)

// Step 2: !(C || D) -> !C && !D
// (x <= 0) || (y <= 10 && z != 5)

🔥 Advanced Examples (Teacher-Level)

Example 1

// Original:
!(a == b && c != d)

// Result:
(a != b || c == d)

Example 2

// Original:
!(temperature > 90 || humidity > 80)

// Result:
(temperature <= 90 && humidity <= 80)

Example 3

// Original:
!(isLoggedIn && hasPermission)

// Result:
(!isLoggedIn || !hasPermission)

Example 4 (AP CSA Trap Question)

// Original:
if(!(num % 2 == 0 || num > 100))

// Rewrite:
if(num % 2 != 0 && num <= 100)

🎓 Why DeMorgan’s Laws Matter (Real Programming)

Simplify complex Boolean expressions
Reduce logical errors
Improve readability
Convert negative logic into positive logic
Answer AP CSA logical reasoning questions

🧪 Common Student Mistakes

❌ Wrong:

!(A && B) == !A && !B

✅ Correct:

!(A && B) == !A || !B

❌ Forgetting to reverse comparison operators

Original	Negated
`>`	`<=`
`<`	`>=`
`==`	`!=`

💥 Advanced Practice (Teacher-Level)

Rewrite using DeMorgan:

```
!(x < 5 && y > 10)
```
```
!(grade >= 90 || extraCredit == true)
```
```
!(loggedIn || isGuest)
```

🧱 Real AP CSA Strategy

When you see:

!( ... && ... )

Change AND → OR
Negate each part

When you see:

!( ... || ... )

Change OR → AND
Negate each part

🔥 Teacher Tip: NOT distributes across parentheses and flips AND/OR.