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Discrete Mathematics with Graph Theory (3rd Edition) 6

Discrete Mathematics with Graph Theory (3rd Edition) 6 - 4...

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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 1.
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