Boolean Algebra

Step 1: Define Essential Prime Implicant (EPI).
An Essential Prime Implicant (EPI) is a prime implicant that covers a 1 that no other prime implicant covers.

Step 2: Explanation of Prime Implicant.
A Prime Implicant (PI) whose each 1 is covered by a minimum of one Essential Prime Implicant (EPI) is termed an Essential Prime Implicant. It means that the coverage of 1's is critical and cannot be omitted by any other implicant.

Step 3: Analysis of options.
- (A) Essential prime implicant: This is correct because the definition of EPI matches this description.
- (B) Selective prime implicant: This is incorrect as this term does not refer to the PI covered by EPIs.
- (C) False prime implicant: This is incorrect as false implicants do not satisfy the required conditions.
- (D) Redundant prime implicant: This is incorrect as redundant implicants are not essential to the minimization process.

Step 4: Conclusion.
The correct answer is (A) Essential prime implicant, as it fits the definition provided in the question.

Step 1: Simplification of Expression.
The given expression is . Observe that , so the expression simplifies to:

Step 2: Using Consensus Theorem.
Now, apply the consensus theorem, which states that . Thus, the simplified expression is:

Step 3: Conclusion.
The output of the minimized expression is , so the correct answer is (B).