sol08 - CS 330 Discrete Computational Structures Fall...

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CS 330 : Discrete Computational Structures Fall Semester, 2014 Assignment #8 Solutions Due Date: Sunday, Nov 2 Suggested Reading: Rosen 9.1 and 9.5, LLM 9.4 These are the problems that you need to turn in. For more practice, you are encouraged to work on the other problems. Always explain your answers and show your reasoning. 1. [20 Pts] Shana For each of these relations on the set of real numbers decide whether it is reflexive, anti-reflexive, symmetric, anti-symmetric and transitive. Justify your answers. (a) ( x, y ) R 1 if and only if xy 0 Solution: reflexive because xx = x 2 0 since squares are always non-negative, for all x . not antireflexive because (0 , 0) R 1 . symmetric because xy = yx , so if ( x, y ) R 1 , then ( y, x ) R 1 . not antisymmetric because (5 , 0) R 1 and (0 , 5) R 1 . not transitive because ( - 1 , 0) R 1 and (0 , 1) R 1 , but ( - 1 , 1) 6∈ R 1 . (b) ( x, y ) R 2 if and only if x 2 y Solution: not reflexive because (1 , 1) 6∈ R 2 . not antireflexive because (0 , 0) R 2 . not symmetric because (2 , 1) R 2 but (1 , 2) 6∈ R 2 . not antisymmetric because ( - 1 , - 2) R 2 and ( - 2 , - 1) R 2 , but - 1 6 = - 2 . not transitive because ( - 4 , - 2) R 2 and ( - 2 , - , 1) R 2 , but ( - 4 , - 1) 6∈ R 2 . 2. [16 Pts] Peter Consider relation R 3 on the set of positive real numbers where ( x, y ) R 3 if and only if x/y ∈ Q . Decide whether it is reflexive, anti-reflexive, symmetric, anti-symmetric and transitive and show that this an equivalence relation. Describe the equivalence classes. What is the equivalence class of 2? of π ? Justify your answers. Solution: Reflexive: For any x R , x x = 1 and 1 Q , so ( x, x ) R 3 for all x . R 3 is reflexive. Symmetric: If ( x, y ) R 3 , then x y Q , so 1 x y = y x Q . Therefore, ( y, x ) R 3 . R 3 is symmetric. Transitive: If ( x, y ) R 3 and ( y, z ) R 3 , then x y Q and y z Q . The product of two rational numbers is rational, so x y y z = x z Q . Therefore, ( x, z ) R 3 . R 3 is transitive. Anti-reflexive: 1 R . 1 1 = 1 Q . Therefore, (1 , 1) R 3 . Since there is at least one element related to itself, R 3 is not anti-reflexive.
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  • Fall '15
  • Soma Chaudhuri
  • Equivalence relation, Rational number, Transitive relation, equivalence class

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