The number of essential prime implicant for the logic expression :

F = ABC + CDA + BD + AC

Explanation:

To determine the number of essential prime implicants for the given logic expression:

F = ABC + CDA + BD + AC

we need to follow the steps of simplifying the Boolean expression using Karnaugh Maps (K-maps) and identifying essential prime implicants.

Step 1: Convert the Boolean expression into minterms

First, we need to express the given Boolean function in terms of minterms. Each product term in the expression represents a minterm in a K-map. Let's identify the minterms for each product term:

• ABC: This term corresponds to the minterm where A=1, B=1, and C=1. Hence, ABC corresponds to the minterm 111 which is m7.
• CDA: This term corresponds to the minterm where C=1, D=1, and A=1. We can rewrite this as ACD. Hence, ACD corresponds to the minterm 1011 which is m11.
• BD: This term corresponds to the minterms where B=1 and D=1. If we consider all possible combinations of A and C, we get two minterms: 1010 (m10) and 1110 (m14).
• AC: This term corresponds to the minterms where A=1 and C=1. If we consider all possible combinations of B and D, we get four minterms: 1001 (m9), 1011 (m11), 1101 (m13), and 1111 (m15).

Combining all the minterms, we get:

F = m7 + m9 + m10 + m11 + m13 + m14 + m15

Step 2: Construct a Karnaugh Map (K-map)

We will use a 4-variable K-map to simplify the Boolean function. The K-map is as follows:

From the K-map, we can identify the groups (or prime implicants) by grouping adjacent 1s:

• Group 1: m9 (1001) and m11 (1011) - Corresponds to AC
• Group 2: m13 (1101) and m15 (1111) - Corresponds to AC
• Group 3: m10 (1010) and m14 (1110) - Corresponds to BD
• Group 4: m7 (0111) - Corresponds to ABC

Step 3: Identify the essential prime implicants

Essential prime implicants are the prime implicants that cover at least one minterm that no other prime implicant covers. From the above groups:

• Group 1 and Group 2 are not essential as they both cover the same minterms (AC).
• Group 3 is essential as it covers m10 and m14 which are not covered by any other group.
• Group 4 is essential as it covers m7 which is not covered by any other group.

Therefore, the essential prime implicants are:

• BD
• ABC

Thus, the number of essential prime implicants is 2.

Correct Option Analysis:

The correct option is:

Option 1: 2

This option correctly identifies the number of essential prime implicants for the given logic expression.