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4
Solutions to Exercises
Exercises
0.2
1.
(a) [BB] Hypothesis:
a
and
b
are positive numbers.
Conclusion:
a
+
b
is positive.
(b) Hypothesis:
T
is a right angled triangle with hypotenuse of length c and the other sides of lengths
a
and
b.
Conclusion:
a
2
+
b
2
=
c
2
•
(c) [BB] Hypothesis:
p
is a prime.
Conclusion:
p
is even.
(d) Hypothesis:
n>
1 is an integer.
Conclusion:
n
is the product of prime numbers.
(e) Hypothesis: A graph is planar.
Conclusion: The chromatic number is 3.
2. (a) [BB]
a
and
b
are positive is sufficient for
a
+
b
to be positive;
a
+
b
is positive is necessary for
a
and
b
to be positive.
(b) A right angled triangle has sides of lengths
a, b,
c, c the hypotenuse, is sufficient for
a
2
+
b
2
=
c
2
;
a
2
+
b
2
=
c
2
is necessary for a right angled triangle to have sides of lengths
a,
b,
c, c the
hypotenuse.
(c) [BB]
p
is a prime is sufficient for
p
to be even;
p
is even is necessary for
p
to be prime.
(d)
n
>
1 an integer is sufficient for
n
to be the product of primes;
n
a product of primes is necessary
for
n
to be an integer bigger than
(e) A graph being planar is sufficient for its chromatic number to be 3. Chromatic number 3 is
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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