Simplify the Boolean function F together with the don't-care conditions in the following function

Given Boolean function:

$$F(A, B, C) = \sum m(0, 1, 2, 3, 7, 8, 9, 12, 13, 15)$$

$$D(A, B, C) = \sum d(4, 5, 6, 10, 11, 14)$$

Objective: Simplify the Boolean function F considering the don't-care conditions in D.

Step 1: Convert to Canonical SOP Form

Expand F and D into canonical sum-of-products (SOP) form:

$$F(A, B, C) = A'B'C' + A'BC' + ABC' + AB'C + AC'B + A'BC + AB'C' + AC'B' + A'B'C + ABC$$

$$D(A, B, C) = A'B'C + A'BC' + ABC + A'B'C' + AB'C' + ABC'$$

Step 2: Combine Terms with Identical Minterms

Combine terms in F and D that have identical minterms and eliminate duplicate minterms:

$$F(A, B, C) = A'B'C' + A'BC' + ABC' + AB'C + AC'B + A'BC + AB'C' + AC'B'$$

$$D(A, B, C) = A'B'C + A'BC' + ABC + A'B'C' + AB'C' + ABC'$$

Step 3: Identify Prime Implicants

Identify prime implicants, which are minimal terms that cannot be further simplified:

$$P_1 = A'B'C$$

$$P_2 = A'BC'$$

$$P_3 = AB'C$$

$$P_4 = AC'B$$

$$P_5 = ABC'$$

Step 4: Construct Prime Implicant Chart

Create a prime implicant chart to determine essential prime implicants:

Minterm	Prime Implicants
0	P_1, P_2
1	P_2, P_3
2	P_2, P_4
3	P_1, P_2, P_3, P_4
7	P_2, P_5
8	P_3, P_5
9	P_2, P_3, P_5
12	P_1, P_2, P_4, P_5
13	P_2, P_3, P_4, P_5
15	P_1, P_2, P_4, P_5

Step 5: Identify Essential Prime Implicants

Essential prime implicants are those that cover all minterms in at least one row of the chart:

$$P_1 = A'B'C$$

$$P_2 = A'BC'$$

Step 6: Simplify Boolean Function

Simplify F using only the essential prime implicants:

$$F(A, B, C) = A'B'C + A'BC'$$

Simplified Boolean Function with Don't-Care Conditions:

$$F(A, B, C) = A'B'C + A'BC'$$

Conclusion:

The Boolean function F has been simplified considering the don't-care conditions in D. The simplified function is expressed in canonical SOP form and contains only the essential prime implicants. This simplified function is more efficient and easier to implement in digital circuits.