Discrete Structures
Semester
062
Major Examination
I
Max
Time
allowed:
01:15
Hours
Name:
I.D.
No.:
Section
I
Total
1
20
I
I
0
1
Question
1
2
Department of Information
&
Computer Science
King Fahd University of Petroleum
&
Minerals
Full Marks
7
4.5
Score
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Define the Method of Indirect Proof.
Problem
1 [0.6+0.4+1+1.5+1.5+2
points]:
(a)
Answer the followings as True (T) or False (F).
Ans:
Since the implication
p
4
q
is equivalent to its contrapositive,
q
+
p,
the
implication
p
can be proved by showing that its contrapositive,
q
+
is true. An
argument of this type is called an indirect proof.
(c)
Give an indirect proof of the theorem "If 3n
+
2 is odd, then n is odd."
Ans
T
T
T
No.
(i)
(ii)
(iii)
4
Ans:
Assume that the conclusion of this implication is false; namely, assume that n is
even. Then n
=
2k for some integer k.
It follows that 3n
+
2
=
3(2k)
+
2
=
6k
+
2
=
2(3k
+
I),
so 3n
+
2 is even (since it is a multiple of 2). Since the negation of the conclusion of
the implication implies that the hypothesis is false, the original implications is true.
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 Spring '08
 Anafi/Quinlan
 Natural number, Modus ponens, Rule of inference

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