Question

QUESTION 1 [TOTAL MARKS:25] A manufacturing process has four sensors labelled W.X, Y. and Z. The system should sound an alarm if any of the following conditions arise: • W, X, Y, Z are not activated at the same time. • X, Y, and Z are not activated and W is activated at the same time. • Wand Y are not activated, and X and Z are activated at the same time. • W, X, and Z are not activated, and Y is activated at the same time. • X and Z are not activated, and W and Y are activated at the same time. [4 Marks] Q 1(a) Present a truth table for this system. Q 1(b) [6 Marks] Derive, using a Karnaugh Map, a minimal sum-of-products function for this system that will sound the alarm when any one of the listed conditions arises. Q1(c) [2 marks] Present a circuit, using only AND and OR gates, implementing the function derived in part (b). Assume that each gate can have any required number of inputs. Q 1(d) [8 Marks] Using De Morgan's Theorem(s), design a circuit, using only NOR gates, that will implement the function derived in Q1(b). Derive a circuit for the same function using only NAND gates Q 1(e) [5 Marks] Prove the following rule of Boolean algebra using any other rules of Boolean algebra: A+ AB = A+B [End of Question 1]

Answer 1

Question 1(a): Construct truth table: Inputs X Y Output (alarm W Z 11

Question 1(b) Function becomes: F(W,X,Y,Z) = m(0,2,5,8,10) Find Boolean expression using k-map. WX YZ 00 01 10 X. Z -W.x.y.z Therefore, F(W.X,YZ) = m(0,2,5,8,10) =X.Z+W.X.Y.Z

Q1(c) IXN Qi(d) Consider the function. F(W.X,Y,Z) = m(0,2,5,8,10) = X-Z+W.X Y Z Apply de morgan's theorem. F(W.X,Y,Z)= m(0,2,5,8,10) =X.Z+W.X.Y.Z = (%.z).(W.X.Y.Z) =(X+2)(W +X+Y+z) =(X +2)+(W+X+Y+Ž) = (X +Z)+(w +X+Y+z)

Draw the circuit using NAND gate. * N D BIN Q1(e) Consider R.H.S term. A+B= A+B(A+A) = A + AB + A B = A(1+B) + A.B = A+ĀB