Discrete Mathematics with Graph Theory (3rd Edition) 41

Discrete Mathematics with Graph Theory (3rd Edition) 41 -...

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Section 2.3 39 10. (a) y x (b) The relation is not reflexive because. for example. (2,2) fj n. It is not tran- sitive because. for example. (2,0) En and (0, 1) E n but (2, 1) fj n. (c) The relation is symmetric since if (x, y) E n. then 1 ~ Ixl + Iyl ~ 2. so 1 ~ Iyl + Ixl ~ 2. so (y,x) E n. It is not arttisymmetric since. for exam- ple; (0,1) E nand (1,0) E n. but ° ¥= 1. 11. (a) [BB] Reflexive: For any set X. we ha~ X ~ X. Not symmetric: Let a, b E S. Then {a} ~ {a, b} but {a, b} <Z {a}. Antisymmetric: If X ~ Yand Y ~ X. then X = Y. Transitive: If X ~ Y and Y ~ Z. then X ~ Z. (b) Not reflexive: For no set X is it true that X ~ X. Not symmetric: As before. . Antisymmetric "vacuously": It is impossible for X ~ Y and Y ~ X. (Recall that an implication is false only when the hypothesis is true and the conclusion is false.) Transitive: As before. . (c) Not reflexive: Since S ¥= 0, there is some element a E S, and so some set X = {a} ¥= 0 E P(S). For this X, however, X n X = X ¥= 0, so (X, X) fj n. Symmetric: If (X, Y) E n, then X n Y = 0, so Y n X = 0. hence, (Y, X) E n.
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