For each of the following circle ALL CORRECT ANSWERS. There may be one, zero or many correct answers.  If none of the answers are correct, circle nothing.  Each possible answer in the multiple choice will be graded separately-credit will be given for correctly circling or not circling each choice. Problems 1-7 are worth 8 points apiece. Problem 8 is worth 6, 9 and 10 are 4 each, 11and 12 are 7 each, and 13 and 14 are each worth 8.

  1.)  Which of the following unsigned binary numbers is equivalent to the decimal number  70?

 Solution: Convert to binary:

70/2 - 35 R0

35/2 - 17 R1

17/2 -8 R1

8/2 - 4 R0

4/2 - 2 R0

2/2 -1 R0

1/2 - 0 R1

1000110 -Correct Answer C

a.) 01000111    *** c) 01000110  e.)  1010111

  b.) 00100101  d.) 0100110    f.)  0111110

 
        2.) The two's complement of 1011101 is:

 Keep 1st 1, change all the other bits- Correct Answer: D

a) 0100010b.) 1100010c.) 0100101

  ***d.) 0100011e.) 0100001f.) 101110   

  3.) The Hex number 3A9.20 is equivalent to which of the following octal numbers:

 Solution:

 Answer d.

e.) 165.110 f.)969.125g.)9A3.02h.)358.02 

  4.) Which of the following are true?

  a.) x + 0 = 0

False, x + 0 = x

b.) x+1=x'

False, x + 1 = 1

c.)y*1 = 0

False, (anything)*1 = (anything)

d.)y*0=y'

False, (anything)*0 = 0  

e.) x+1 = 1 TRUE  

f.)xy+0 = xy TRUE, x+0 = x

  g.)y*0 = 1

False, (anything)*0 = 0

  h.)y*0 = y'

False, (anything)*0 = 0

CORRECT ANSWER: e & f

  5.) Which of the following are NOT true?

  a.) x + x'y = x+y

True, x+x'y = (x+x')(x+y) = 1*(x+y) = x+y

  b.)x(x'+y) = xy

True x(x'+y) = xx' + xy = 0 + xy = xy

  c.)(x+y+z)(x+y'+z)=(x+z)

True, [(x+z)+y][(x+z)+y']=(x+z)+yy'=x+z+0=x+z

  d.) xy'+y = 1 FALSE

  e.)zx+z = x+z FALSE

  f.)xy'+xy'z = xy'

True

  g.) (x')'= x + xy +xz

True

  h.) xy +(x'+y')z=xy+z

True 

  CORRECT ANSWERS: d & e

  6.) Which of the following expressions are true given that:

F(w,x,y,z)=wx + x'yz

CORRECT ANSWERS: a,c,e

b.) F'= P(0,1,2,4,5,6,7,8,9,10)

***c.) F'= S(0,1,2,4,5,6,7,8,9,10)

d.) F= P(0,1,3,4,5,7,8,9,12,13,15)

***e.) F= S(3,11,12,13,14,15)

g.) F'= S(0,1,3,4,5,7,8,9,12,13,15)

 
7.) Which of the following expressions correspond to the Boolean Function F(x,y,z)= S(1,4,6,7)?

  a.) F=xy'z' + x'yz' + x'y'z + x'y'z'

  b.)F= P(1,4,6,7)

***c.)F=x'y'z+xy'z'+xy

  ***d.)F= P(0,2,3,5)

e.)F=(xy'z'+x'yz'+x'y'z+x'y'z')'

f.)F=(x+y'+z)(x'+y+z)(x'+y'+z)(x'+y'+z')

 
CORRECT ANSWERS: C & D

 

The one on the far right.  The first diagram implements
(XY)' + X'Y.   The middle one is (x'+y)(x+y')

 
 SOLUTION:   First, let's represent each value as an 8-bit binary
 				 two's-complement signed number:
     17:   00010001  (There is only ONE way to represent a positive
number in any format).
    +37:   00100101  To store this as a negative number, take the
2's complement:
    -37:   11011011
Now, perform the addition:

-37= 11011011

  Sum= 11101100

 
To convert back to decimal, realize that this is a negative number
(because the leftmost bit is a 1), so to get the magnitude we must
take the 2's complement:
-(00010100) 

     14.) Using Boolean algebra, expand the following equation to a Sum of minterms (i.e., canonical SOP form). Next, using Boolean Algebra, reduce the equation to a minimum number of literals.

F= a'b' + c' + a'bc

 
	SOLUTION:  To get to Sum of minterms, we'll expand the function 
	algebraically:

F= a'b'(c+c') + (a+a')(b+b')c' + a'bc 
 = a'b'c' + a'b'c + a'b'c' + a'bc' + ab'c' + abc' + a'bc
   000      001     000      010     100     110    011
Sorting and getting rid of duplicates:

 
Next, we'll simplify the original function algebraically:
  a'b' + a'bc + c' = a'(b'+bc) + c' = a'(b'+b)(b'+c) + c' 
  = a'(b'+c) + c' = a'b' + a'c + c' = a'b' + (a'+c')(c+c')
  = a'b' + a' + c' 
  = a' + c' (absorption theorem)