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Discrete Mathematics with Graph Theory (3rd Edition) 3

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

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Solutions to Exercises Exercises 0.1 1. (a) [BB] True (e) True (b) [BB] Not valid (f) Not valid (c) [BB] False (0 2 is not positive.) (g) False 2. (a) [BB] True, because both 4 = 2 + 2 and 7 < J50 are true statements. (b) False, because one of the two statements is false. (c) [BB] False, because 5 is not even. (d) True, because 16- 1 / 4 = ~. (e) [BB] True, since 9 = 3 2 is true (or because 3.14 < 71'). (f) True, because (_4)2 = 16 is true. (g) [BB] True, because both hypothesis and conclusion are true. (h) False, because the hypothesis is true but the conclusion is false. (i) [BB] Not a valid mathematical statement. (j) True, because both statements are true. (k) True, because this is an implication with false hypothesis. (1) False, because one of the statements is false while the other is true. (m) False, because the hypothesis is true but the conclusion is false. (d) Not valid (n) [BB] False, because the area of a circle of radius r is not 271'r and its circumference is not
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Unformatted text preview: 71'r2. (0) False, because the hypothesis of this implicatio~ is true, but the conclusion is false. (p) [BB] This is true: The hypothesis is true only when a ~ band b ~ a, that is, when a = b, and then the conclusion is also true. (q) This implication is true because the hypothesis is always false. 1 3. (a) [BB] If x > 0, then -> O. x (b) If a and b are rational numbers, then ab is a rational number. (c) If f is a differentiable function, then f is conti~uous. (d) [BB] If 9 is a graph, then the sum of the degrees of the vertices of 9 is even. (e) [BB] If A is a matrix and A # 0, then A is invertible. (f) If P is a parallelogram, then the diagonals of P bisect each other. (g) If n is an even integer, then n < O. (h) If two vectors are orthogonal, then their dot product is O. (i) If n is an integer, then n~l is not an integer. (j) If nis a natural number, then n + 3> 2. 4. (a) [BB] True (the hypothesis is false). (b) True (hypothesis and conclusion are both true). 1...
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