Discrete Mathematics with Graph Theory (3rd Edition) 49

Discrete - Section 2.5 47(c This is not a partial order because the relation is not antisyrrunetric for example 3 j 3 because_3)2 ~ 32 a nd 3 j 3

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Section 2.5 47 (c) This is not a partial order because the relation is not antisyrrunetric; for example, -3 j 3 because (_3)2 ~ 3 2 and 3 j -3 because 3 2 ~ (_3)2 but -3 ::j:. 3. (d) This is not a partial order because the relation is not antisyrrunetric; for example, (1,4) j (1,8) because 1 ~ 1 and similarly, (1,8) j (1,4), but (1,4) ::j:. (1,8). (e) This is a partial order. Reflexive: For any (a, b) E N x N, (a, b) j (a, b) because a ~ a and b ~ b. Antisymmetric: If (a, b), (c, d) E N x N, (a, b) j (c, d) and (c, d) j (a, b), then a ~ c, b ~ d, c ~ a and d ~ b. So a = c, b = d and, hence, (a, b) = (c, d). Transitive: If (a, b), (c, d), (e, f) E N x N, (a, b) j (c, d) and (c, d) j (e, f), then a ~ c, b ~ d, c ~ e and d ~ f. So a ~ e (because a ~ c ~ e) and b ~ f (because b ~ d ~ f) and, therefore, (a, b) j (e, f). This is not a total order; for example, (1,4) and (2,5) are incomparable. (t) This is reflexive and transitive but not antisymmetric and, hence, not a partial order. For example,
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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