HOMEWORK 1

EE/CS – 220 DIGITAL LOGIC CIRCUITS Deadline: Monday, 14 February 2011 Q1. Assume that there is a large room with three doors and that a switch near each door controls a light in the room. You can turn the light on or off by changing the state of the switches.

a) Draw a truth table that depicts the above situation. b) Simplify the function using K-map c) Draw a circuit using minimum number of logic gates that implements the function

obtained in part b. Q2. In most computer systems it is often required to select a certain input from a number of input signals to a system. Suppose there are two data input signals (X and Y) and a selection control input (S) to a system. If S=0 anything that is on input X would appear on the output and if S=1 the data on the input Y would appear on the output.

a) Draw a truth table that depicts the above situation. b) Simplify the function using K-map c) Draw a circuit using minimum number of logic gates that implements the function

obtained in part b. Q3. Build a circuit with three inputs (x1, x2 and x3) that controls a given digital system. The circuit should recognize three different conditions:

Condition A is true if x3 is true and either x1 is true or x2 is false Condition B is true if x1 is true and either x2 or x3 is false Condition C is true if x2 is true and either x1 is true or x3 is false

The control circuit must produce a 1 if atleast two of the conditions A, B and C are true. Design the simplest circuit that can be used for this purpose. Q4. Design a circuit with output f and inputs x1, x0, y1 and y0. Let X = x1x0 be a number where the four possible values of X, namely, 00, 01, 10 and 11 represents the four numbers 0, 1, 2 and 3 respectively. Similarly, let Y=y1y0 represent another number with the same possible values. The output f should be 1 if the numbers represented are equal. Otherwise, f should be 0.

a) Show the truth table for f b) Synthesize the simplest possible product of sums expression for f

Q5. Repeat problem 4 for the case where f should be 1 only if X≥Y.

a) Show the truth table for f b) Show the canonical sum-of-products expression for f c) Show the simplest possible sum-of-products expression for f

Q6. Design the simplest possible circuit that has three inputs x1, x2 and x3, which produces and output value of 1 whenever exactly one or two variables have the value 1; otherwise, the output has to be 0. Q7. Simplify using K-map

1) E(w,x,y,z) = Σ(0,7,8,10,13) 2) F(a,b,c,d) = ∏(1,3,4,6,8,9,11,12,14) 3) G(x2,x1,x0) = Σ(0,1,5,7,8,10,14,15)

Q8. Find the minterms of the following expressions by first plotting each expression on a map:

1) BD + AB + BD

2) YWYZWXYYXWXZ Q9. Team D is about to qualify for the next round, however it’s qualification depends on the results from the last game of the teams (A, B, C) in its group. The team qualifies if: 1) If both A and C lose their last game or 2) A wins, and both B and C lose their last game or 3) C loses and B wins their respective last games Write a Boolean expression for the qualifying condition of D. Note: Variable A = 1 if Team A loses and A = 0 if Team A wins Q10. Simplify the following expressions using Boolean algebra.

i. ))()(0()))(()(0()( CBBADCBACBBADCBA

ii. DACDAABA ))0()1.((

iii. DCACABAAB )(

iv. )))((()))((( BABABABA

v. )( CBBCBA

vi. )()( DBCBADBBCA

Q11. Convert the following to sum of product and product of sums.

i. ))(( EFBCDCBA

ii. ))(( ZYYXXX

Q12. Use DeMorgan’s theorem and convert the following

i. CDBCACBA into product of sums

ii. ))()(( DCBADCADBA into sum of products