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CMSC 250

Quiz #11

Monday, Nov. 19, 2001

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the quiz is 30, and you will have exactly 15 minutes to complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.

1. [10 pnts.] Let $A = \{2,3,4,5\}$ and let $B = \{4,5,6,7,8\}$. Let $S:A \rightarrow B$ be defined by
   
   $$\forall (a,b) \in A \times B, (a,b) \in S \leftrightarrow a \mbox{ divides } b$$
   
   
   Explicitly state the elements of $S$.
   
   
   
   
   
   
   
2. [10 pnts.] Indicate with a ``YES'' or ``NO'' in each blank if the following relations are Reflexive, Symmetric, and/or Transitive. Assume all are relations over the set $A=\{2,4,6\}$.
   1. $R_1 = \{(2,2),(2,4),(2,6),(4,4),(4,2),(6,6),(6,2)\}$
      1. Reflexive
      2. Symmetric
      3. Transitive
   2. $R_2 = \{(2,2),(2,4),(2,6),(4,6)\}$
      1. Reflexive
      2. Symmetric
      3. Transitive
   3. $R_3 = \{(2,2),(2,4),(4,2),(4,4)\}$
      1. Reflexive
      2. Symmetric
      3. Transitive
3. [10 pnts.] Let $M = \{1,2,3,4,5\}$ and let $R: M \rightarrow M$ be defined as
   
   $$R = \{(1,3),(2,2),(3,1),(3,2),(3,3),(1,1),(4,5)\}$$
   
   
   Give the list of the elements in the set R' (the transitive closure of R).

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