Math 272 Test 1 Fall 1995

Dr. P. D. Williams, Dept. of Mathematics, CSUSB

Attempt all questions. Marks for each question are given in brackets. Show all necessary work for full or partial credit. You may use a calculator.

1. Let , and . Find , and . [8]

   Solution

   .
   , .

2. Write out a truth table for the statement . [8]

   Solution

   

3. Use truth tables to determine if and are equivalent. Write out English statements for both of these if p, q and r are the statements ``a is related to b", ``b is related to c" and ``a is related to c", respectively. [20]

   Solution

   

   If a is related to b and b is related to c then a is related to c. If a is not related to c, then a is not related to b or b is not related to c (or both).

4. Determine whether one of the following pair of statements implies the other: [12]

   

   Solution

   

   Since neither is true when the other is true, there is no implication.

5. A relation R from to is defined by aRb if and only if a+b is even. Write down this relation as a subset of . Draw a digraph of this relation. [12]

   Solution

   

   .

   

6. Draw a Venn diagram to illustrate why the distributive law

   

   holds in a universal set U. [10]

   Solution

   

   corresponds to regions 1-6 and so corresponds to regions 1, 2 and 4. The set corresponds to regions 1 and 2 while corresponds to 1 and 4. Thus the union corresponds to regions 1,2 and 4 as required.

7. Use Boolean algebra to show that . You may use the fact that . [10]

   Solution

   

   using the associative law and the given fact.

8. Write down the relation corresponding to the digraph below. Write down the adjacency list for this digraph. Determine the vertices reachable from vertex a. Finally, explain why this is, or is not, a transitive relation. [20]

   Solution

   Not available on the net as yet

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