Consider a logic circuit shown in figure below. The functions f1, f2 and f (in canonical sum of products form in decimal notation) are:
f1(w,x,y,z) = sum(8,9,10)
f2(w,x,y,z) = sum(7,8,12,13,14,15)
f(w,x,y,z) = sum(8,9)
The circuit has f1 and f2 as inputs to an AND gate, and f3 as a second input to an OR gate whose output is f. That is: f = (f1 AND f2) OR f3.
The function f3 is
A. sum(9,10)
B. sum(9)
C. sum(1,8,9)
D. sum(8,10,15)
GATE 1997 · Digital Logic · Circuit Output · medium
Answer: f3 = sum(9). Answer: B.
Compute f1 AND f2 (intersection of minterm sets): f1 minterms: {8, 9, 10}
f2 minterms: {7, 8, 12, 13, 14, 15}
Intersection: {8}
So f1 AND f2 = sum(8)
Solve for f3 using f = (f1 AND f2) OR f3: We need: sum(8) OR f3 = sum(8,9)
f3 must contain minterm 9 (to add it to the output).
f3 must NOT contain any minterm outside {8,9}, otherwise f would have extra minterms.
Minimal solution: f3 = sum(9).
Check option A: f3=sum(9,10) -> sum(8) OR sum(9,10) = sum(8,9,10) != sum(8,9). FAIL.
Check option B: f3=sum(9) -> sum(8) OR sum(9) = sum(8,9) = f. PASS.
Check option C: f3=sum(1,8,9) -> sum(8) OR sum(1,8,9) = sum(1,8,9) != sum(8,9). FAIL.
Check option D: f3=sum(8,10,15) -> sum(8) OR sum(8,10,15) = sum(8,10,15) != sum(8,9). FAIL.