🔲 Boolean Logic¶

The foundation of digital circuits and programming conditions

What You'll Learn¶

Logic gates: AND, OR, NOT, and more
Truth tables
Boolean algebra and De Morgan's Laws
How boolean logic powers programming

What is Boolean Logic?¶

Boolean logic is a branch of algebra that deals with true and false values (1 and 0). It was invented by George Boole in the 1800s and is the foundation of: - Computer circuits - Programming conditions (if statements) - Database queries - Search engines

Think of boolean logic as the "decision-making" system of computers.

The Three Basic Gates¶

AND Gate¶

Output is TRUE only if ALL inputs are TRUE.

A ──┐
    ├─── Output
B ──┘

A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

Real-world: A car that needs both a key AND the brake pressed to start.

OR Gate¶

Output is TRUE if ANY input is TRUE.

A ──┐
    ├─── Output
B ──┘

A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

Real-world: A light switch that can be turned on from upstairs OR downstairs.

NOT Gate (Inverter)¶

Output is the OPPOSITE of the input.

A ──[NOT]─── Output

A	NOT A
0	1
1	0

Real-world: A light that turns on when it gets DARK (NOT light).

More Logic Gates¶

NAND (NOT AND)¶

Output is FALSE only when ALL inputs are TRUE.

A	B	A NAND B
0	0	1
0	1	1
1	0	1
1	1	0

Fun fact: NAND gates are "universal" - you can build ANY other gate using only NANDs!

NOR (NOT OR)¶

Output is TRUE only when ALL inputs are FALSE.

A	B	A NOR B
0	0	1
0	1	0
1	0	0
1	1	0

XOR (Exclusive OR)¶

Output is TRUE when inputs are DIFFERENT.

A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

Real-world: A light that toggles when either switch is flipped.

XNOR (Exclusive NOR)¶

Output is TRUE when inputs are the SAME.

A	B	A XNOR B
0	0	1
0	1	0
1	0	0
1	1	1

Logic Gate Reference¶

Gate	Symbol	Python	Description
AND	∧	`and`	True if both are true
OR	∨	`or`	True if either is true
NOT	¬	`not`	Inverts the value
NAND	↑	`not (a and b)`	Not AND
NOR	↓	`not (a or b)`	Not OR
XOR	⊕	`a != b`	True if different
XNOR	⊙	`a == b`	True if same

Boolean Algebra¶

Boolean algebra defines rules for working with boolean values.

Basic Laws¶

Identity Laws:

A AND 1 = A
A OR 0 = A

Null Laws:

A AND 0 = 0
A OR 1 = 1

Idempotent Laws:

A AND A = A
A OR A = A

Complement Laws:

A AND (NOT A) = 0
A OR (NOT A) = 1

Double Negation:

NOT (NOT A) = A

Commutative, Associative, Distributive¶

Commutative:

A AND B = B AND A
A OR B = B OR A

Associative:

(A AND B) AND C = A AND (B AND C)
(A OR B) OR C = A OR (B OR C)

Distributive:

A AND (B OR C) = (A AND B) OR (A AND C)
A OR (B AND C) = (A OR B) AND (A OR C)

De Morgan's Laws¶

De Morgan's Laws are powerful tools for simplifying boolean expressions.

Law 1¶

NOT (A AND B) = (NOT A) OR (NOT B)

In words: "Not (A and B)" means "Not A or Not B"

Example: - "Not (sunny and warm)" = "Not sunny OR Not warm"

Law 2¶

NOT (A OR B) = (NOT A) AND (NOT B)

In words: "Not (A or B)" means "Not A and Not B"

Example: - "Not (rainy or cold)" = "Not rainy AND Not cold"

Boolean Logic in Programming¶

If Statements¶

# AND
if has_ticket and is_adult:
    enter_concert()

# OR
if is_weekend or is_holiday:
    sleep_in()

# NOT
if not is_raining:
    go_for_walk()

# Combined
if (has_key and has_permission) or is_admin:
    open_door()

Truthiness in Python¶

Python treats certain values as "falsy": - False - 0 (integer) - 0.0 (float) - "" (empty string) - [] (empty list) - {} (empty dict) - None

Everything else is "truthy".

name = input("Enter your name: ")
if name:  # True if name is not empty
    print(f"Hello, {name}!")
else:
    print("You didn't enter a name.")

Short-Circuit Evaluation¶

Python only evaluates what it needs to:

# AND: Stops at first False
False and expensive_function()  # expensive_function never runs!

# OR: Stops at first True
True or expensive_function()    # expensive_function never runs!

Building Complex Gates¶

You can combine basic gates to create complex logic:

Half Adder¶

Adds two single bits, outputs sum and carry.

A ──┬──[XOR]─── Sum
    │
B ──┴──[AND]─── Carry

Full Adder¶

Adds three bits (including carry from previous addition).

Multiplexer (MUX)¶

Selects one of many inputs based on a selector.

Common Mistakes ⚠️¶

1. Confusing `and`/`or` precedence¶

# Wrong
if age > 18 and has_id or is_vip:
    # Parsed as: (age > 18 and has_id) or is_vip
    pass

# Better with parentheses
if (age > 18 and has_id) or is_vip:
    pass

2. Using `==` instead of `=` in conditions¶

# Wrong - assignment, not comparison
if x = 5:  # SyntaxError!
    pass

# Right
if x == 5:
    pass

3. Double negatives¶

# Confusing
if not (not is_valid):
    pass

# Simpler
if is_valid:
    pass

4. Overcomplicating conditions¶

# Overcomplicated
if (x > 0 and x < 10) or (x > 0 and x == 5):
    pass

# Simplified
if x > 0 and x < 10:
    pass

Try It Out! 🚀¶

Run examples.py to see boolean logic in action:

python examples.py

Then try exercises.py to practice:

python exercises.py

Key Takeaways¶

AND is true only when all inputs are true
OR is true when any input is true
NOT inverts the value
XOR is true when inputs are different
De Morgan's Laws help simplify complex conditions
Boolean logic is the foundation of all programming decisions

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