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Discrete Mathematics with Graph Theory (3rd Edition) 40

Discrete Mathematics with Graph Theory (3rd Edition) 40 -...

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38 Solutions to Exercises Not antisymmetric: (2,1) En because 2 -1 = 1 \$ 3 and (1, 2) En because 1- 2 = -1 \$ 3, but 1"# 2. Not transitive: (5,3) E n because 5 - 3 = 2 \$ 3 and (3,1) E n because 3 - 1 = 2 \$ 3, but (5,1) sf n because 5 - 1 = 41. 3. (t) Reflexive: For any (a, b) E A, a - a = b - b; thus, «a, b), (a, b)) En. Symmetric: If «a, b), (c, d)) En, then a-c = b-d, so c-a = d-b and, hence, «c, d), (a, b)) E n. Notantisymmetric: «5,2), (15, 12)) En because 5-15 = 2-12 and similarly, «15,12), (5,2)) E n; however, (15,12) "# (5,2). If «a, b), (c,d)) E nand «c,d), (e,f)) E n then a - c = b - d and c - e = d - I. Thus, a - e = (a - c) + (c - e) = (b - d) + (d - f) = b - I and so «a, b), (e, f)) E n. (g) Not reflexive: If n E N, then n "# n is not true. Symmetric: Ifnl "# n2, then n2 "# nl. Not antisymmetric: 1 "# 2 and 2"# 1 so both (1,2) and (2, 1) are in n, yet 1 "# 2. Not transitive: 1 "# 2, 2 "# 1, but 1 = 1. (h) Not reflexive: (2,2) sf n because 2 + 2 "# 10. Symmetric: If (x, y) E n, then x + y = 10, so y + x = 10, and hence, (y, x) En. Not antisymmetric: (6, 4) E n because 6 + 4 = 10 and similarly,
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