2.1.1. Commutative Laws

a. A ∨ B = B ∨ A
The commutative law for the “or” operation (∨) indicates that the result of the operation is independent of the order of the variables. Thus, A ∨ B and B ∨ A yield the same result.

b. A ∧ B = B ∧ A
The same rule applies for the “and” operation (∧). A ∧ B is equivalent to B ∧ A.

2.1.2. Associative Laws

a. A ∨ (B ∨ C) = (A ∨ B) ∨ C
The associative law for the “or” operation shows that the order in which the operations are performed does not affect the result. Grouping B and C first and then combining with A yields the same result as grouping A and B first and then combining with C.

b. A ∧ (B ∧ C) = (A ∧ B) ∧ C
The same holds true for the “and” operation; the order of operations does not impact the outcome.

2.1.3. Distributive Laws

a. A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
This law expresses the distribution of the “and” operation over the “or” operation. The intersection of A with B ∨ C is equivalent to the union of the intersections of A with B and A with C individually.

b. A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
This law shows the distribution of the “or” operation over the “and” operation. The union of A with B ∧ C is equivalent to the intersection of A with B and A with C individually.

2.1.4. Identity Laws

a. A ∨ 0 = A
The “or” operation between a variable and 0 (FALSE) results in the variable itself. In Boolean algebra, 0 is considered “FALSE” or the “empty set.”

b. A ∧ 1 = A
The “and” operation between a variable and 1 (TRUE) results in the variable itself. In Boolean algebra, 1 represents “TRUE” or the “universal set.”

2.1.5. Complement Laws

a. A ∨ A′ = 1
The “or” operation between a variable and its complement (negation) always yields 1 (TRUE), encompassing all possible outcomes.

b. A ∧ A′ = 0
The “and” operation between a variable and its complement always results in 0 (FALSE), corresponding to the empty set in set theory.

2.1.6. Idempotent Laws

a. A ∨ A = A
The “or” operation between a variable and itself results in the variable.

b. A ∧ A = A
The “and” operation between a variable and itself results in the variable.

These laws illustrate how Boolean algebra structures logical operations and simplifies expressions, aiding in the reduction and analysis of complex logical expressions.