Exercise 4 - Tonga Institute of Higher Education

TONGA INSTITUTE OF HIGHER EDUCATION
IT 235 – Discrete Mathematics - SEMESTER 2: 2012
Revision Exercises No. 4
: Construct a truth table for each of these compound statements.
.
%.
'.
(.
.
+.
p q" # p $ q"
p $ q" # p & q"
p q" $ p & q"
p ) q" $ ~p ) q"
p ) q" $ ~p ) ~r"
p $ q" # p $ ~q"
,: Which of the following are propositions? What are the truth values of those that
are propositions.
. Miami is the capital of Florida
%. There are no black 7lies in 8Eua
' Answer the following questions
(. What time is it?
. x < y > y < x for any pair of real numbers a and y
+. 4 < x > 5
A. x < 1 > 5 if x > 1
C: Write each of the following in the form p # q in English. Hint: Refer to the
list of common ways to express implications"
. It rains whenever the wind blows from the north
%. The roses will bloom if it stays warm for a week
'. If you drive more than 400 kilometers, you will need to buy gasoline.
(. I will remember to send you the address only if you send me an email message.
. You get promoted only if you have connections and you have connections only if you
get promoted.
+. It is necessary to have a valid password to log on to the server.
J: Find the bitwise OR , bitwise AND and the bitwise XOR for the following pairs
bitwise strings
a. 101 1100 1101, 111 0111 1011
b. 1001 0011 1111, 1110 1100 1111
c. 100 1111 0111, 110 0000 1111
d. 1111 0111 0011, 0101 1111 0111
e. 1010 1111 0101 1100, 1110 1100 1010 011
Question 5. Show that the following implications are tautologies without using truth tables.
. O~p & p
(. Op
q"P # q
%. Op # q" & q # r"P # p # r"
'. Op & p # q"P # q
q" & p # r" & q # r"P # r
Question 6. Use truth table to verify the Absorption Laws:
. p
p & q" Q p
%. p & p
q" Q p
Question 7. Show that the following pairs of compound propositions are logically equivalent
using truth tables and also without using truth tables.
. p ) q RST p & q"
U p &U q"
%. U p ) q" RST U p ) q
'. p # r" & q # r" and p
(. p # q"
q" # r
p # r" and p # q
r"
Question 8. Find the dual of each of the following propositions.
a. p & ~q & ~r
b. p & q & r"
s
c. p
V" & q & W"
End of Exercise 4

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