1) (15pts.) Perform the following addition, writing your answer in binary. Convert each of the operands to decimal and hexadecimal.
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1 1 0
1 1 1 0 decimal value: 110 hexadecimal: 6E
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1 0 0 0 0 1 0 1 |
2) (12 pts.) (Show your work for credit.)
a)
Convert 50110 to octal. 7658
8|501
8|62 r 5.
8|7 r 6
0 r 7
b)
Determine the decimal value of the following Java bytes (two’s
complement rules apply).
0101 1010
26 + 24 + 23 + 21
= 90
1010 1110
0101 0010
26 + 24 + 21 = -82
3)
(10 pts.) Find the expression, S, representing the following
truth table by writing out the disjunctive normal form of S
(do not simplify) then draw the circuit:
S = pqr + pq’r’ + p’q’r’
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p |
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q |
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r |
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S |
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T |
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T |
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T |
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T |
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T |
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T |
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F |
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F |
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T |
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F |
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T |
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F |
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T |
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F |
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F |
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T |
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F |
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T |
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T |
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F |
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F |
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T |
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F |
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F |
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F |
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F |
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T |
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F |
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F |
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F |
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F |
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T |
4)
(12 pts.) Minimize the following expression using either
Boolean Algebra or Karnaugh map (show your work for credit): You should use the other method to check
your results.
F = p’q’r’ + p’qr’ + pq’r’ + pqr + pqr’ = r’ + pq
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p\qr |
00 |
01 |
11 |
10 |
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0 |
1 |
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1 |
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1 |
1 |
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1 |
1 |
5) (12 pts.) Given the conditional statement: p Ú r ® ~p.
a)
Write this expression in its alternate representation using
the symbols: ~ and Ú
~(p Ú r) Ú
~p
b)
Using DeMorgan’s law and others (Theorem 1.1.1), simplify the
expression
~(~(p Ú
q) Ú
~p)
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~((~p Ú ~q) Ú
~p) |
Alternate answer after same first step |
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~(~p) |
(p Ú q) Ù
p |
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p |
p |
c) Use the expression in its original form: p Ú r ® ~p
i) Write the contrapositive
p ® ~(p Ú
r) Alternate answer:
p ® ~p Ù
~r
ii) Write the inverse (of the original)
~(p Ú r) ®
p Alternate answer: ~p Ù
~r ® p
6)
(7 pts.) Either verify or disprove the following argument form
using truth tables (justify your conclusion):
p ® q Ù p
~p Ú q ® p
\~p
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p |
q |
p ® q Ù p |
~p Ú q ® p |
CR |
~p |
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T |
T |
T |
T |
Ö |
F |
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T |
F |
F |
T |
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F |
T |
T |
F |
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F |
F |
T |
F |
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Conclusion: The argument form is valid/invalid
(choose one), because: Conclusion does not follow
from the premises.
7) (8 pts.) Consider the statement: "(x, y) Î D, $ y such that P(x,y) Ú Q(y)
a) Write the negation of the
statement.
$ x, y Î D such that " y ~P(x, y) Ù ~Q(y)
Alternate answer: $ x, y Î D such that " y ~(P(x, y) Ú Q(y))
b) Simplify by applying alternate representation for the conditional
and Demorgan’s Law:
"m,n
Î
Z,
if n > 0 or m > 0, then
max(n, m) ³
0.
"m, n Î Z (n < 0 Ù m < 0) Ú max(n, m) > 0.
8)
(9 pts.) For each of the following argument forms, identify it
as Universal Modus Ponens, Universal Modus Tollens, converse error or inverse
error.
a) b)
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Any sum of two rational numbers is rational. Converse Error c) |
For all students x, if x studies discrete
mathematics, then x is good at logic. UMP |
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If a number is even, then twice that number is even. |
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9)
(15 pts.) The following logic puzzles were written by Lewis
Carroll, author of Alice’s Adventures in Wonderland. Write a symbolic
representation of the given hypothesis then find the best conclusion (a
sentence) that can be formed by using all the hypotheses. If no conclusion can
be formed, then state none.
Hypotheses Symbolic
Representation
a)
No son of mine is dishonest; S
® H
People always treat an honest man with respect. H ® R
\
People respect my sons.
b)
All cats understand French; C
® F
Some chickens are cats. $ Ch ®
C
\Some chickens understand French
c)
All diligent students are successful; D ®
S
All ignorant students are unsuccessful. I ® ~S
\Diligent students are not ignorant
d)
Babies are illogical; B ®
I
Nobody is despised who can handle a crocodile; C ® ~D
Illogical persons are despised; I ® D
\Babies can’t handle crocodiles.
Crocodile handlers are not babies.
Babies are despised.
Any one of the above for full credit.
