Boolean Algebra Laws

Why This Matters

Boolean algebra isn't just abstract symbol manipulation—it's the mathematical foundation underlying everything from digital circuit design to database queries to the logic gates in your computer's processor. When you're tested on these laws, you're being asked to demonstrate that you understand how logical operations behave and why certain transformations preserve truth values. The laws you'll learn here show up constantly in proof simplification, circuit optimization, and formal verification problems.

The key insight is that Boolean algebra has a beautiful dual structure: most laws come in pairs where AND and OR mirror each other. Understanding this duality means you can essentially learn half the laws and derive the rest. Don't just memorize formulas—know which law to apply when simplifying an expression, and recognize when two expressions are logically equivalent. That's what separates students who struggle with proofs from those who breeze through them.

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Laws of Structural Rearrangement

These laws tell you that the structure of how you write an expression doesn't affect its truth value—you can reorder and regroup without changing meaning.

Commutative Laws

• Order doesn't matter—you can swap the position of operands freely: $A \land B = B \land A$ and $A \lor B = B \lor A$
• Applies to both AND and OR—this mirrors how addition and multiplication work with numbers, making Boolean algebra feel algebraically familiar
• Exam application: When simplifying expressions, rearrange terms to group similar variables together before applying other laws

Associative Laws

• Grouping doesn't matter—parentheses can be shifted without changing the result: $(A \land B) \land C = A \land (B \land C)$
• Works for both operations—similarly, $(A \lor B) \lor C = A \lor (B \lor C)$, letting you flatten nested expressions
• Practical use: Allows you to write $A \land B \land C$ without ambiguity, since all groupings are equivalent

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Laws of Expansion and Simplification

These laws let you transform expressions between expanded and simplified forms—essential for proofs and circuit optimization.

Distributive Laws

• AND distributes over OR—$A \land (B \lor C) = (A \land B) \lor (A \land C)$, similar to how multiplication distributes over addition
• OR distributes over AND—$A \lor (B \land C) = (A \lor B) \land (A \lor C)$, which is unlike regular algebra (this is unique to Boolean algebra!)
• Critical for simplification: Use distribution to factor common terms or expand expressions for comparison

Absorption Laws

• A variable absorbs expressions containing itself—$A \land (A \lor B) = A$ and $A \lor (A \land B) = A$
• Intuition: If $A$ is true, the whole expression reduces to $A$; if $A$ is false, the expression also evaluates based solely on $A$
• Simplification power: Look for absorption opportunities when a variable appears both inside and outside parentheses

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Laws of Identity and Annihilation

These laws describe what happens when variables meet the constants 0 (false) and 1 (true)—the boundary conditions of Boolean algebra.

Identity Laws

• 1 is the identity for AND—$A \land 1 = A$, since AND-ing with true doesn't restrict the outcome
• 0 is the identity for OR—$A \lor 0 = A$, since OR-ing with false adds no new truth conditions
• Dual structure: Notice how 1 and 0 swap roles between AND and OR—this is the duality principle in action

Idempotent Laws

• Repeating a variable is redundant—$A \land A = A$ and $A \lor A = A$
• Named from Latin idem potens—meaning "same power," indicating the operation has no additional effect
• Simplification use: Eliminate duplicate terms that arise from substitution or expansion

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Laws of Negation and Complements

These laws govern how negation interacts with variables and operations—crucial for transforming expressions and understanding logical opposites.

Complement Laws

• A variable AND its negation yields false—$A \land \lnot A = 0$, since something can't be both true and false
• A variable OR its negation yields true—$A \lor \lnot A = 1$, representing the law of excluded middle
• Proof strategy: Recognizing complement patterns lets you immediately simplify to constants

Double Negation Law

• Negating twice returns the original—$\lnot(\lnot A) = A$
• Intuitive meaning: "Not not-A" is just A; two negations cancel out
• Common in proofs: Often appears after applying De Morgan's Laws, requiring cleanup

De Morgan's Laws

• Negation of AND becomes OR of negations—$\lnot(A \land B) = \lnot A \lor \lnot B$
• Negation of OR becomes AND of negations—$\lnot(A \lor B) = \lnot A \land \lnot B$
• Essential transformation: These laws let you "push" negation inside parentheses while swapping the operation—critical for normal form conversions

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The Duality Principle

This isn't a law you apply directly—it's a meta-principle that explains the elegant symmetry of Boolean algebra.

Duality Principle

• Every theorem has a dual—swap AND with OR, and swap 0 with 1, and you get another valid theorem
• Example: The dual of $A \land 1 = A$ is $A \lor 0 = A$—both are identity laws
• Learning efficiency: Once you prove or memorize one law, you automatically know its dual is also true

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Quick Reference Table

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Self-Check Questions

1. Which two laws both involve rearranging terms without changing truth value, and how do they differ in what they rearrange?

2. You encounter the expression $\lnot(P \lor Q)$. Which law do you apply, and what is the result?

3. Compare and contrast the Identity Laws and the Complement Laws—both involve constants, so what distinguishes when each applies?

4. An expression contains $A \land (A \lor B)$. Which law simplifies this most directly, and what does it simplify to?

5. Using the Duality Principle, what is the dual of De Morgan's Law $\lnot(A \land B) = \lnot A \lor \lnot B$? Verify that it's a valid Boolean law.