3. (10 pt.) R is a relation on Z such that (x, y) E R if and only if there is a

Question

3. (10 pt.) R is a relation on Z such that (x, y) E R if and only if there is a positive integer rn such that rn -y. Determine whether R is a partial order, a strict order, or an equivalence relation. Justify your answers. 4. (20 pt., 10 pt. each) Draw a Hasse diagram for each of the following partial orders. Find the maximal and minimal elements and determine whether the partial order is a total order. a. R is a partial order on the set [0, 1,2,3,4, 5) such that (x.y) e R if and only if x2 y. b. R is a partial order on the set {2,3,5,10,11,15,25] such that (x,y) ER if and only if y is divisible by x. Hint: An integer y is divisible by an integer x with x # 0 if and only if there exists an integer k such that y = xk.

Explanation / Answer

Partial order (reflexive,antisymmetric,transitive)
Strict order (Irreflexive,asymmetric,transitive)
Equivlence order(reflexive, symmetric, transitive)

3.   x^n = y (xRy)

   it is irreflexive as aRa does not hold
   It is asymmetric as when aRb does then bRa does not hold
   it is transitive aRb and bRc implies aRc

   It is strict order

4 b) y is divisible by x (xRy)
     It is a partial order

     it is reflexive as aRa holds
     it is antisymmetric as aRb and bRa implies a = b
     it is tansitive as aRb and bRc implies aRc
   

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