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CMSC 250

Quiz #12

Monday, Dec. 3, 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. [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.
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
   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.
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. Reflexive
      2. Symmetric
      3. Transitive
      4. Irreflexive
      5. Antisymmetric
      6. 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. Reflexive
      2. Symmetric
      3. Transitive
      4. Irreflexive
      5. Antisymmetric
      6. Asymmetric

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