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Discrete Mathematics with Graph Theory (3rd Edition) 5

Discrete Mathematics with Graph Theory (3rd Edition) 5 -...

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Section 0.1 (i) Converse: If p( x) is a polynomial with at least one real root, then p( x) has odd degree. Contrapositive: If p( x) is a polynomial with no real roots, then p( x) has even degree. (j) Converse: A set of at most n vectors is linearly independent. Contrapositive: A set of more than n vectors is not linearly independent. 3 (k) Converse: If f is not one-to-one, then, for all real numbers x and y, x '" y and x 2 + xy + y2 + x+y = O. Contrapositive: If f is one-to-one, then there exist real numbers x and y such that x = y or x 2 + xy + y2 + X + y '" O. (1) [BB] Converse: If f is not one-to-one, then there exist real numbers x and y with x '" y and x 2 + xy + y2 + X + y = O. Contrapositive: If f is one-to-one, then for all real numbers x and y either x = y or x 2 + xy + y2 +x+y = O. 7. (a) [BB] There exists a continuous function which is not differentiable. (b) 2 X 2: 0 for allreal numbers x. (c) [BB] For every real number x, there exists a real number y such that y > x. (d) For every set of primes PI , P2, ... ,Pn, there exists a prime not in this set.
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