CMSC 250

Quiz #12 KEY

Monday, Dec. 3, 2001

1. [18 pnts.] For each of the following, give the new relation requested. Make sure to use the representation specified to show the relation.
   1. Let R be a relation on the set $A = \{1,2,3,4,5\}$
      where R is defined as $R = \{(1,1),(1,4),(2,3),(2,4),(2,2),(3,1),(3,3),(4,1),(4,4)\}$.
      Give the transitive closure of R using the MATRIX notation.
      
      
      
      ANSWER:
      
      
      $$\begin{array}{r\vert\vert ccccc\vert} &1&2&3&4&5\\ \hli... ... 3&1&0&1&1&0\\ 4&1&0&0&1&0\\ 5&0&0&0&0&0\\ \end{array}$$
      
      
      
      GRADING:
      9 points possible
      -2 for each 1 or 0 out of place to max of -9
      or -4 if correct relation given, but did used digraph or set notation form
      (more off in addition to the -3 for any incorrect entries in the relation)
      The labels I have put here on the rows and columns are not necessary.
      
      
      
      
      
   2. Let P be a relation on the set $B = \{1,2,3\}$ where the Matrix representation of the relation P is
      
      $$M_P = \begin{array}{\vert ccc\vert} 1&1&1\\ 0&0&0\\ 1&1&0\\ \end{array}$$
      
      
      Give the matrix representation of X where X is the symmetric closure of P.
      
      
      
      ANSWER:
      
      
      $$M_X = \begin{array}{r\vert\vert ccc\vert} &1&2&3\\ \hline \hline 1&1&1&1\\ 2&1&0&1\\ 3&1&1&0\\ \end{array}$$
      
      
      
      GRADING:
      9 points possible
      -2 for each 1 or 0 out of place to max of -9
      or -4 if correct relation given, but did used digraph or set notation form
      (more off in addition to the -3 for any incorrect entries in the relation)
      
      
      
      
      
2. [12 pnts.] Write a ``YES'' or ``NO'' in each blank to indicate if the relation described has the listed property.
   1. Let M be a relation on the set $A=\{1,2,3,4\}$
      where $M = \{(1,1),(1,2),(2,2),(2,3),(3,3),(4,1),(1,4),(4,4)\}$.
      1. YES Reflexive
      2. NO Symmetric
      3. NO Transitive
      4. NO Irreflexive
      5. NO Antisymmetric
      6. NO Asymmetric
   2. Let N be a relation on the set of $\Z$ where $Z = \{(x,y) \in \Z \times \Z \vert x = y + 3 \mbox{ or } x = y - 2\}$
      1. NO Reflexive
      2. NO Symmetric
      3. NO Transitive
      4. YES Irreflexive
      5. YES Antisymmetric
      6. YES Asymmetric
      GRADING:
      1 point per blank (12 blanks - 12 points)

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