Boolean Identities for AND Operation:
The AND operation is indicated by (.) in the Boolean algebra. Following are the Boolean identities pertaining to AND operation.
A . B = B . A (Commutative Law)
This implies that the inputs of the AND gate may be interchanged without changing the output Y that may be justified from the truth table of two-input AND gate. Note down that (.) is suppressed several times and we may write the Commutative law as A B = B A.
A B C = (A B) C = A (B C) (Associative Law)
This means that the order of combining the input variables contain no effect on the output variables. This may be verified from the truth table for three-input AND gate.
A A = A
This means that any variable ANDed with itself equals the variable. For A = 0, 0 . 0 = 0 and for A = 1, 1 . 1 = 1 is true (refer to truth table for two-input AND gate).
A . 1 = A
If one input of the AND gate is high the output is equivalent to the input. For A = 0, 0.1 = 0 and for A = 1, 1.1 = 1 is true.
A . 0 = 0
If one input of the AND gate is low the output is low irrespective of the other input. For A = 0, 0.0 = 0 and for A = 1, 1.0 = 0 is true.
A (B + C) = A B + A C (Distributive Law)
We can write the truth tables for LHS and RHS of the Boolean equation and check out they are same.