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 $\mathrm{\neg}$ (logical NOT), satisfying specific axioms.
Basic Operations:

Logical AND ($\wedge $):
 Represents conjunction.
 $A\wedge B$ is true only when both $A$ and $B$ are true.

Logical OR ($\vee $):
 Represents disjunction.
 $A\vee B$ is true when at least one of $A$ or $B$ is true.

Logical NOT ($\mathrm{\neg}$):
 Represents negation.
 $\mathrm{\neg}A$ 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: $(A\wedge B)\wedge C=A\wedge (B\wedge C)$, $(A\vee B)\vee C=A\vee (B\vee C)$
 Distributive Law: $A\wedge (B\vee C)=(A\wedge B)\vee (A\wedge C)$, $A\vee (B\wedge C)=(A\vee B)\wedge (A\vee C)$
 Complement Law: $A\wedge \mathrm{\neg}A=0$, $A\vee \mathrm{\neg}A=1$
 Absorption Law: $A\wedge (A\vee B)=A$, $A\vee (A\wedge B)=A$
 De Morgan's Law: $\mathrm{\neg}(A\wedge B)=\mathrm{\neg}A\vee \mathrm{\neg}B$, $\mathrm{\neg}(A\vee B)=\mathrm{\neg}A\wedge \mathrm{\neg}B$
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.