Given below is the truth table for a combinational circuit for which the input is a 3 bit number and output is its 2's complement.

Inputs			Outputs
X	Y	Z	P	Q	R
0	0	0	0	0	0
0	0	1	1	1	1
0	1	0	1	1	0
0	1	1	1	0	1
1	0	0	1	0	0
1	0	1	0	1	1
1	1	0	0	1	0
1	1	1	0	0	1

Write SOP expression for the outputs P, Q and R. Reduce them, if possible, using the Karnaugh's map.

SOP for P:
X'Y'Z + X'YZ' + X'YZ + XY'Z'

SOP for Q:
X'Y'Z + X'YZ' + XY'Z + XYZ'

SOP for R:
X'Y'Z + X'YZ + XY'Z + XYZ

K-Map for P:

From Pair (1,3):
Rows representing the Pair: X'
Columns representing the Pair: Y'Z + YZ = Z
Term Obtained = X'Z

From Pair (3,2):
Rows representing the Pair: X'
Columns representing the Pair: YZ + YZ' = Y
Term Obtained = X'Y

From (4):
Rows representing (4): X
Columns representing (4): Y'Z'
Term Obtained = XY'Z'

Reduced Expression for P = XY'Z' + X'Y + X'Z

K-Map for Q:

From Pair (1,5):
Rows representing the Pair: X' + X = 1
Columns representing the Pair: Y'Z
Term Obtained = Y'Z

From Pair (2,6):
Rows representing the Pair: X' + X = 1
Columns representing the Pair: YZ'
Term Obtained = YZ'

Reduced Expression for Q = Y'Z + YZ'

K-Map for R:

From Quad (1,3,5,7):
Rows representing the Quad: X' + X = 1
Columns representing the Quad: Y'Z + YZ = Z
Term Obtained = Z

Reduced Expression for R = Z