Boolean Algebra

1. Logic Operators

In Boolean algebra, every variable and operator returns a binary value of either $0$ or $1$.

Basic Operators

These are the foundational building blocks of all logical circuits.

Set Operator Title	Algebraic Expression	Logical Representation	Logic Gate Symbol
Intersection	$A\cdot B$	AND
Union	$A+B$	OR
Complementary	$A^{\prime }$ or $\overline{A}$	NOT
Identity	$A$	BUFFER

Derived Operators

These operators are combinations of the basic operators, often used to simplify circuit design.

Set Operator Title	Algebraic Expression	Logical Representation	Logic Gate Symbol
Alternative Denial	$(A\cdot B{)}^{\prime }$	NAND
Joint Denial	$(A+B{)}^{\prime }$	NOR
Symmetric Difference	$A\oplus B$	XOR
Equivalence	$(A\oplus B{)}^{\prime }$	XNOR

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2. Axioms and Theorems

Boolean logic is defined by a set of axioms (assumed truths) and theorems (proven rules). Every rule has a Dual, which is equally valid.

Axioms

The fundamental assumptions of the binary field.

Name	Axiom	Dual
Binary Field	$B=0$ if $B\ne 1$	$B=1$ if $B\ne 0$
NOT	$\overline{0}=1$	$\overline{1}=0$
AND / OR	$0\cdot 0=0$	$1+1=1$
AND / OR	$1\cdot 1=1$	$0+0=0$
AND / OR	$0\cdot 1=1\cdot 0=0$	$1+0=0+1=1$

Theorems

Rules used for simplifying Boolean expressions.

Name	Theorem	Dual
Identity	$B\cdot 1=B$	$B+0=B$
Null Element	$B\cdot 0=0$	$B+1=1$
Idempotency	$B\cdot B=B$	$B+B=B$
Involution	$\overline{\overline{B}}=B$	$\overline{\overline{B}}=B$
Complements	$B\cdot \overline{B}=0$	$B+\overline{B}=1$
Commutativity	$B\cdot C=C\cdot B$	$B+C=C+B$
Associativity	$(B\cdot C)\cdot D=B\cdot (C\cdot D)$	$(B+C)+D=B+(C+D)$
Distributing	$(B\cdot C)+(B\cdot D)=B\cdot (C+D)$	$(B+C)\cdot (B+D)=B+(C\cdot D)$
Covering	$B\cdot (B+C)=B$	$B+(B\cdot C)=B$
Combining	$(B\cdot C)+(B\cdot \overline{C})=B$	$(B+C)\cdot (B+\overline{C})=B$
Consensus	$(B\cdot C)+(\overline{B}\cdot D)+(C\cdot D)=(B\cdot C)+(\overline{B}\cdot D)$	$(B+C)\cdot (\overline{B}+D)\cdot (C+D)=(B+C)\cdot (\overline{B}+D)$
De Morgan’s	$\overline{B_0\cdot B_1\cdot \dots }=(\overline{B_0}+\overline{B_1}+\dots )$	$\overline{B_0+B_1+\dots }=(\overline{B_0}\cdot \overline{B_1}\cdot \dots )$

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3. Boolean Duality

Duality is a central property of Boolean algebra. A dual expression is derived by replacing:

• $\cdot$ (AND) with $+$ (OR)
• $+$ (OR) with $\cdot$ (AND)
• $0$ with $1$
• $1$ with $0$

Generalized Duality Principle:

$f(X_1,X_2,\dots ,X_n,0,1,+,\cdot )⟺f(X_1,X_2,\dots ,X_n,1,0,\cdot ,+)$

NOTE

The Duality Principle states that any theorem that can be proven is automatically proven for its dual.

Warning

Duality is not the same as De Morgan’s Law. Duality swaps operators and constants but does not complement the individual variables.