Discrete Mathematics For AIML - Chapter 6 Boolean Variables & Functions

Chapter 6 — Boolean Algebra & Digital Logic

This chapter covers Boolean algebra and digital logic, essential for representing logical relationships, rule-based AI systems, and digital circuit simulations in AI & ML.

6.1 Boolean Variables & Functions

Boolean variables take values True (1) or False (0). Boolean functions combine variables using logical operations. AI/ML Context: Representing features, decision rules, and binary classifiers.

# Example in Python
A = True
B = False

def boolean_function(A, B):
    return A and not B

print(boolean_function(A, B))  # Output: True

6.2 Boolean Operations

AND (∧): True if both inputs are True
OR (∨): True if at least one input is True
NOT (¬): Inverts the input
XOR (⊕): True if inputs are different

# Python examples
A = True
B = False

print(A and B)   # AND -> False
print(A or B)    # OR -> True
print(not A)     # NOT -> False
print(A != B)    # XOR -> True

6.3 Simplification Using Boolean Laws

Boolean expressions can be simplified using laws for efficient computation and circuit design.

Identity Law: A ∧ 1 = A, A ∨ 0 = A
Complement Law: A ∧ ¬A = 0, A ∨ ¬A = 1
Distributive Law: A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
De Morgan’s Law: ¬(A ∧ B) = ¬A ∨ ¬B, ¬(A ∨ B) = ¬A ∧ ¬B

# Python example of De Morgan's Law
A = True
B = False

print(not (A and B))   # True
print((not A) or (not B))  # True

6.4 Karnaugh Maps (K-Maps)

K-Maps provide a visual method to simplify Boolean expressions. AI/ML Context: Simplifying logical rules and feature interactions in decision systems.


# Example: 2-variable K-Map
# Variables: A, B
# Function: F(A,B) = Σ(1,3)
# K-Map simplifies F = B

6.5 ML Use Case: Logical Feature Interactions & Rule-based AI

Binary classification using Boolean features
Rule-based expert systems
Logical preprocessing for feature engineering
Simulation of digital circuits or logic gates in AI models

6.6 Practical Python Applications

# Example: Logical rule system
features = {"is_raining": True, "have_umbrella": False}

def should_go_out(features):
    return features["have_umbrella"] or not features["is_raining"]

print(should_go_out(features))  # Output: False

6.7 Why Boolean Algebra & Digital Logic Matter in AI/ML

Represent logical rules for decision-making systems.
Simplify binary feature interactions.
Simulate digital circuits and logic layers in AI models.
Foundational knowledge for advanced topics like knowledge graphs and symbolic AI.

6.8 Exercises

Simplify Boolean expressions using De Morgan’s laws.
Implement a small rule-based AI system in Python.
Create a K-map for a 3-variable Boolean function and simplify it.

Hints / Solutions

Use logical equivalences to reduce expressions step by step.
Python and, or, not operators can implement rules.
K-map groups of 1s help visualize simplification patterns.

6.9 Further Reading & Videos

Boolean Algebra and Digital Logic by Morris Mano
Python logical operations tutorials (Corey Schafer, YouTube)
Karnaugh Map simplification guides on GeeksforGeeks
Rule-based AI systems and expert system research papers

Next Chapter: Matrices & Linear Transformations — introducing matrix operations, vector spaces, and their applications in AI & ML.

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