1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another theorem in the same system d) Is the main building block of all computers, e) None of the above. 1.2 Sum of products in principle: a) Uses more OR gates than AND gates, b) Uses more AND gates than OR gates, c) Uses XOR gates in combination with , d) None of the above. 1.3 Multiplexer: a) Has usually more inputs than outputs, b) Has usually more outputs than inputs, c) Can be used to implement logic functions in a convenient way, d) Cannot be implemented using standard basic logic gates as NOR, XOR, NAND, e) None of the above. 1.4 An axiom: a) Can easily be proved by induction, b) Is a logical statement that is assumed to be true, c) Can be contradicted by another axiom in the same system d) Is the main building block of all computers, e) None of the above 1.5 Product of sums in principle: a) Uses more OR gates than AND gates, b) Uses more AND gates than OR gates, c) Uses XOR gates in combination with inverters, d) None of the above

1.6 Ripple-carry adder: a) Gives simultaneously the result of addition of all bits, b) Adds MSB first and LSB last when the carry is propagated, c) Adds LSB first and MSB last when the carry is propagated, d) Adds bits of binary numbers disregarding carry, e) None of the above 1.7 A Karnaugh map for a function of six (6) variables would have: a) 8 cells, b) 16 cells, c) 32 cells, d) 64 cells, e) 128 cells. 1.8 A priority encoder with 4 inputs has: a) 2 outputs, b) 3 outputs, c) 4 outputs, d) 16 outputs. 1.9 The product of maxterms f x, y, z, w  M 0,1,4,7 can be written as: b) b)

f x, y, z, w   m2,3,5,6 ,

f x, y, z, w   m0,1,4,7 ,

f x, y, z, w   m2,3,5,6 ‘,

c) d) 1.10

f x, y, z, w  M 2,3,5,6' .

a) b) c) d)

Which of the following is an axiom of Boolean algebra: x . x = x, 1 + 1 = 1, x . x’ = 0, x + x’ = 1.

a) b) c) d)

A multiplexer with 4 select inputs has: 1 output, 2 outputs, 4 outputs, 16 outputs.

1.11

1.12 a) b) c) d)

Which of the following Boolean equations is in Product-Of-Sums (POS) form? F = (A’ B C) (A C’) (A B C) F = A’ B C + A (B’ + C) F = (A’ + B + C’) (A’ + C) (A + B’ + C) F = A B’ C + A’ C’ + A’ B C’

2. Answer in your own words 2.1 What does DeMorgans theorem state? 2.2 What does Shannon’s expansion theorem state? 2.3 What is a DECODER? 2.4 What is a MULTIPLEXER / DEMULTIPLEXER? 2.5 Analyze the CMOS circuit shown in the figure! Fill the truth table and state what logical function does it implement? Indicate pull-up and push-down networks! A

B

Ci

0

0

0

0

0

1

0

1

0

0

1

1

1

0

0

1

0

1

1

1

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1

1

2.6 Write the expression for the function f(a,b) and make the truth-table of the circuit!

Out

2.7 What are the outputs Y in the figures below?!

2.8 What is terribly wrong in the figure below?!

2.9 What is the output F in the figure below?!

2.10

What is a maxterm?

2.11

What is a minterm?

2.12

Determine by perfect induction whether the following is true or false A' + A B = A' + B

2.13

What is the simplest Boolean expression for the circuit shown below

2.14 Using DeMorgan's theorem and Boolean algebra determine whether the two circuits shown here are equivalent or not

3. Solve the following problems 3.1 Make a truth table of function F(x0, x1, x2, x3) that has output logic “1” when inputs form binary numbers divisible by three, and logic “0” otherwise. Construct the reduced circuit using K-map to optimize the expression for F(x0, x1, x2, x3)! 3.2 A minority circuit is one whose output is equal to 1 if the input variables have less 1’s than 0’s. Show the truth table for a 3-input version of this circuit. Write the expression for the reduced circuit using K-map to optimize F(x, y, z)! 3.3 Make a truth table of function F(x0, x1, x2, x3) that has output logic “1” when inputs form a prime binary number, and logic “0” otherwise. Construct the reduced circuit using K-map to optimize the expression for F(x0, x1, x2, x3)! 3.4 Using Karnaugh map, find a minimal sum-of-products expression for the following function: F(x,y,z)=(m1m2m4m5m6)

3.5 Using 2-to-1 MULTIPLEXER, implement the function from previous question. 3.6 Covert the decimal number 187 into binary and hexadecimal numbers! 3.7 Use Karnaugh Map to simplify the following Boolean function F A,B,C, = (0,2,3,10,11,1 ,1 ) into a) sum-of-products form b) product-of-sums form 3.8 Using 8-to-1 MULTIPLEXER and some logical gates, implement the function from question 3.2 3.9 Implement the following Boolean function with 4x1 multiplexer and external gates F A,B,C,

=

(0,1,2,

, , 11)

3.10 Determine the outputs A and B as functions of inputs x, y and z. Can you tell what device is implemented using the multiplexers (for BONUS points)? x

y

z

0

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1

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1

1

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1

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1

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1

1

4. Number Systems, Logic gates and Boolean algebra 4.1 Simplify the following Boolean function to a minimum number of literals:

A

B

a) b) c) d)

A' B’ C’ ’ + A B' CD + B’ C ’ + B C’ D + BCD ((A’ B’)’ + (C’ ’)’ )’ + A + ’ X X + Y’ Y’ + X’ Y’ + X’ Y (AB)’ + (A’ + B) (B’ + B) (A’ + B’)

4.2 Convert the following binary numbers to the indicated bases a) 10111011 to octal b) 1001011011101 to hexadecimal c) 11000101.101 to decimal 4.3 Perform subtraction using 2’s complement of the subtrahend: 01100111 – 00101011 4.4 Simplify the following Boolean function to a minimum number of literals: x’y’z+(x+y’)(xz)’ 4.5 Implement the function from Problem 4.4 using ONLY NAND and inverter gates, 4.6 Covert the decimal number 246 into binary and hexadecimal numbers! 4.7 Covert the number 145(6) into a decimal number! 4.8 Perform subtraction using 2’s complement of the subtrahend: 1101001 – 0110011 4.9 Covert the hexadecimal number 187 into binary and decimal numbers! 4.10

Find the base X if 101(X) = ??(10) = 62(8)

4.11

Complete the following table (except the shaded cells) of equivalent values

Binary

Octal

Decimal

011010.11 65 206

Hexadecimal