Q.Convert Y = $A\overline{B} + BC + B\overline{D}$ into its canonical SOP form and write the truth table for the corresponding min term designation.

Each product term of the four-variable function is expanded to include all four variables A, B, C, D, and the resulting minterms are listed with a truth table. …

[!TLDR]

Expanding every term to four variables gives $Y=\Sigma m(4,6,7,8,9,10,11,12,14,15)$.

Variables: A, B, C, D (weights 8 4 2 1).

Step 1 — expand each term to the missing variables:

• $A\overline{B} = A\overline{B}(C+\overline{C})(D+\overline{D})$ → minterms 8 (1000), 9 (1001), 10 (1010), 11 (1011).
• $BC = (A+\overline{A})BC(D+\overline{D})$ → minterms 6 (0110), 7 (0111), 14 (1110), 15 (1111).
• $B\overline{D} = (A+\overline{A})B(C+\overline{C})\overline{D}$ → minterms 4 (0100), 6 (0110), 12 (1100), 14 (1110).

Step 2 — collect the distinct minterms: {8,9,10,11} ∪ {6,7,14,15} ∪ {4,6,12,14} = {4, 6, 7, 8, 9, 10, 11, 12, 14, 15}. Hence $Y=\Sigma m(4,6,7,8,9,10,11,12,14,15)$.

Step 3 — truth table (Y = 1 for the minterms above):