Discrete Mathematics with Graph Theory (3rd Edition) 13

# Discrete Mathematics with Graph Theory (3rd Edition) 13 - =...

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Section 1.1 11 16. Let the rational numbers be ~ and~. We may al!sume that a, b, e, d are positive integers and that a C Th d b Th hi th g±£' bad C d tho . th . a g±£ b < d' us a < e. e nt suggests at b+d IS etween band' an IS IS e case. b < b+d is equivalent to a(b + d) < b(a + e) and m < ~ is equivalent to (a + e)d < (b + d)e, both of which are true because ad < be. 17. On a standard checker board, there are 32 squares of one color and 32 of another. Since squares in opposite corners have the same color, the hint shows that our defective board has 32 squares of one color and 30 of the other. Since each domino covers one square of each color, the result follows. 18. (a) We leave the primality checking of 1(1), ... ,1(39) to the reader, but note that
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Unformatted text preview: = 412. (b) f(k 2 + 40) = 40 2 + 80k 2 + k4 + 40+ k 2 + 41 = k4 + 81k 2 + 412 = (k 2 + 41)2 -k 2 = (k 2 + 41 + k)(k 2 + 41 -k). 19. The answer is no, since 333333331 = 19607843 x 17. Exercises 1.1 1. (a) [BB] p q --.q (--.q) V P p/\((--.q)Vp) T T F T T T F T T T F T F F F F F T T F (b) p q --.p (--.p) -+ q p/\q (p/\q) V ((&amp;quot;:&amp;quot;p) -+ q) T T F T T T T F F T F T F T T T F T F F T F F F (c) p q qVp p/\ (qV p) --.(p/\(qVp)) --. (p/\ (qV p)) +-t P T T T T F F T F T T F F F T T F T F F F F F T F (d) [BB] p q r --.q pV (--.q) --. (p V (--.q)) --.p (--.p) V r (--. (p V (--.q))) /\ ((--.p) V r) T T T F T F F T F T F T T T F F T F F T T F F T T T T F F T T T F T T F T T F F T F F F F T F F T T F F F F F T F F F T T T T F F F T T F T T F...
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