Boolean Algebra --- Core Rules (Simple & Practical)

Boolean algebra is a system for working with logical values:

• 1 / true
• 0 / false

It is used everywhere in programming: if statements, filtering, bit operations, SQL conditions, hardware logic, etc.

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Basic Operators

Operator	Symbol	Meaning	Scientific Naming (Discrete Math / Logic)
AND	a && b or a ∧ b	true if both are true	Conjunction
OR	a || b or a ∨ b	true if at least one is true	Disjunction
NOT	!a or ¬a	logical negation	Negation

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Core Laws

1️⃣ Idempotent Law

Repeating the same value changes nothing.

a && a = a
a || a = a

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2️⃣ Commutative Law

Order does not matter.

a && b = b && a
a || b = b || a

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3️⃣ Associative Law

Grouping does not matter.

(a && b) && c = a && (b && c)
(a || b) || c = a || (b || c)

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4️⃣ Distributive Law (Very Important)

a && (b || c) = (a && b) || (a && c)
a || (b && c) = (a || b) && (a || c)

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5️⃣ Identity (Neutral Elements)

a && true  = a
a || false = a

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6️⃣ Domination (Absorbing Elements)

a && false = false
a || true  = true

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7️⃣ Complement Law

a && !a = false
a || !a = true

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8️⃣ Double Negation

!!a = a

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9️⃣ De Morgan's Laws (Extremely Important)

!(a && b) = !a || !b
!(a || b) = !a && !b

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🔟 Absorption Law

a || (a && b) = a
a && (a || b) = a

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Example Simplification

!(a && b)

Using De Morgan:

!a || !b

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What Is Used Most in Programming

In real-world code, these rules are used most often:

• De Morgan's Laws
• Double negation (!!a)
• Complement law (a && !a)
• Distributive law
• Absorption law

These are especially common when simplifying:

• complex if conditions\
• SQL WHERE clauses\
• filtering logic\
• guard conditions\
• bitwise operations

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Boolean algebra is not just theory --- it is the foundation of clean condition design.