# Boolean Algebra

Definition:

• Boolean algebra is a mathematical structure that deals with binary variables and logical operations. It is a set of elements together with two operations, typically denoted as $\wedge$ (logical AND), $\vee$ (logical OR), and $\mathrm{¬}$ (logical NOT), satisfying specific axioms.

Basic Operations:

1. Logical AND ($\wedge$):

• Represents conjunction.
• $A\wedge B$ is true only when both $A$ and $B$ are true.
2. Logical OR ($\vee$):

• Represents disjunction.
• $A\vee B$ is true when at least one of $A$ or $B$ is true.
3. Logical NOT ($\mathrm{¬}$):

• Represents negation.
• $\mathrm{¬}A$ is true when $A$ is false, and vice versa.

Key Concepts:

• Boolean Variables: Represented by $0$ (false) and $1$ (true).

• Truth Tables: Tables that show the output of a logical operation for all possible input combinations.

• Laws of Boolean Algebra:

• Idempotent Law: $A\wedge A=A$, $A\vee A=A$
• Commutative Law: $A\wedge B=B\wedge A$, $A\vee B=B\vee A$
• Associative Law: $\left(A\wedge B\right)\wedge C=A\wedge \left(B\wedge C\right)$, $\left(A\vee B\right)\vee C=A\vee \left(B\vee C\right)$
• Distributive Law: $A\wedge \left(B\vee C\right)=\left(A\wedge B\right)\vee \left(A\wedge C\right)$, $A\vee \left(B\wedge C\right)=\left(A\vee B\right)\wedge \left(A\vee C\right)$
• Complement Law: $A\wedge \mathrm{¬}A=0$, $A\vee \mathrm{¬}A=1$
• Absorption Law: $A\wedge \left(A\vee B\right)=A$, $A\vee \left(A\wedge B\right)=A$
• De Morgan's Law: $\mathrm{¬}\left(A\wedge B\right)=\mathrm{¬}A\vee \mathrm{¬}B$, $\mathrm{¬}\left(A\vee B\right)=\mathrm{¬}A\wedge \mathrm{¬}B$

Applications:

• Logic Gates: Fundamental building blocks in digital circuits (AND, OR, NOT gates) are based on Boolean algebra.

• Digital Electronics: Used in designing and analyzing digital circuits, computing systems, and hardware.

• Computer Science: Boolean algebra forms the basis of Boolean logic used in programming and computer architecture.

Boolean Functions:

• Boolean Expression: An expression formed using Boolean variables and operations.

• Canonical Form: A form of Boolean expression where each term corresponds to a specific combination of variables that makes the function true.

• Simplification: The process of reducing Boolean expressions to their simplest form using algebraic rules.