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Boolean Algebra Postulates

The fundamental axioms defining the algebraic structure of Boolean variables (0 and 1). These include the identity, complement, and absorption laws, forming the basis for digital logic operations.
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The statement of the theorem

Let (B,∧,∨,¬)(\text{B}, \land, \lor, \neg) be the Boolean algebra defined over the set B={0,1}\text{B} = \{0, 1\}. The structure must satisfy the following axioms for all A,B∈BA, B \in \text{B}: \n1. Commutativity: A∧B=B∧AA \land B = B \land A and A∨B=B∨AA \lor B = B \lor A.\n2. Associativity: (A∧B)∧C=A∧(B∧C)(A \land B) \land C = A \land (B \land C) and (A∨B)∨C=A∨(B∨C)(A \lor B) \lor C = A \lor (B \lor C).\n3. Distributivity: A∧(B∨C)=(A∧B)∨(A∧C)A \land (B \lor C) = (A \land B) \lor (A \land C) and A∨(B∧C)=(A∨B)∧(A∨C)A \lor (B \land C) = (A \lor B) \land (A \lor C).\n4. Identity: A∧1=AA \land 1 = A and A∨0=AA \lor 0 = A.\n5. Complement: A∧¬A=0A \land \neg A = 0 and A∨¬A=1A \lor \neg A = 1.\n6. Absorption: A∧(A∨B)=AA \land (A \lor B) = A and A∨(A∧B)=AA \lor (A \land B) = A.