Question

Prove the following identity using algebraic manipulation:
ABC+ABC′+A′B=B

Answer 1

To prove the identity ABC + ABC' + A'B = B, we use Boolean algebra principles, applying the distributive laws and the fact that A + A' always equals 1 and AA' always equals 0, ultimately showing the expression simplifies to B.

Step-by-step explanation:

Proving the Algebraic Identity

To prove the identity ABC + ABC' + A'B = B, let's use algebraic manipulation and Boolean algebra principles. Remember, the prime (') denotes logical NOT. Here's the step-by-step explanation:

1. ABC + ABC' = AB(C + C') = AB(1) = AB because C + C' is always 1 in Boolean algebra.
2. Similarly, A'B can be rewritten as AB' using the identity B + B' = 1, giving A'BB' which simplifies to 0B' as AA' is always 0.
3. Now, combine AB and A'B to get B(A + A') = B(1) = B, because A + A' is 1. The entire expression simplifies to B, which proves the given identity.

Through these steps, the use of commutative and distributive laws, and understanding of how 0s and 1s work in Boolean algebra, we confirmed the identity.