Math 105            Practice  Exam 2   (with answers)     Name ____________________________

Do your work on this paper.  You may add additional sheets.  Harder questions are marked with a *.

1.   Circle the letter in front of any of the following that are statements in logic.
  
  a. Does Math 105 meet on Wednesdays?
               b. All horses are brown.                                               b and d
               c. Blueberry pie tastes better than apple pie.
               d. Cornell is in Ithaca.

2.  Write down a negation of each of the following statements without putting something like “it is not true that” in front of the statement.  

      a.   1994 was not a leap year.     1994 was a leap year.

      b.   Some houses are white.      All houses are not white.

      c.    A triangle has four sides.     A triangle does not have four sides.

       d.   Bob or Kim attended the concert.  [Hint: Use one of De Morgan’s Laws]

                        Neither Bob nor Kim attended the conference.

       e.   All students missed Math 105 class on 4/5/07.   Some students went to Math 105 on 4/5/07.

      f.    If   is a rational number then e is not a rational number.  

                      is a rational number and e is a rational number.

3.      Let p, q, and r represent the statements:
     
      p: The light is on.
      q: The picture is not visible.
      r:  The sun has set.

  Express the following in English :

        a.  ~pq       The light is not on and the picture is not visible.
        b.  (~rq)  ~p    If the sun has not set or the picture is not visible, then the light is not on.
 
 Express the following in symbolic form:

       c.   The picture is visible if and only if the light is on or the sun has set.  ~q  pr

       d.   The light is on does not imply that the picture is visible.  ~(p~q)  or   pq

4. Circle the quantified statement that is equivalent to the given statement.
     
     All dogs like bones. 

     a.   Some dogs do not like bones. 
     b.   Not all dogs dislike bones. 
     c.   All cats like bones and all dogs like bones. 
     d.   No dog dislikes bones.                                               d

5.   Given the following statement, write its converse, inverse and contrapositive and circle the appropriate True or False for each.

      T    F       If the world is flat, then 4 is an even number.     true

      T    F       Converse:  If 4 is an even number then the world is flat.   false

      T    F        Inverse:    If the world is not flat, then 4 is not an even number   false

      T    F       Contrapositive:  If 4 is not an even number then the world not is flat.   true       

6.  Fill in the following truth tables.

  p 	  q	 ~p	~pq    	
  				
  T	  T	 F	T	
  T	  F	 F  	T	
  F	  T	 T  	T	
  F	  F	 T	F	
                                   
p	q	r	  ~q	 p~q	(p~q)r	
			   			
T	T	T	F	F	T	
T	T	F	F	F	F	
T	F	T	T	T	T	
T	F	F	T	T	T	
F	T	T	F	F	T	
F	T	F	F	F	F	
F	F	T	T	F	T	
F	F	F	T	F	F	

7.  Use a truth table to determine if  q  pq  is logically equivalent to ~p  q.

   They are not logically equivalent.  If you do the truth table you find column for one is TTFT and the other is TFTT.

.

8.  For each of the following, draw a valid conclusion from the given premises.

       If all  students pass the course, then no students repeat the course.
       Some  students repeat the course..

Therefore: Some students failed the cours.   

        If  I work in the garden,  my back gets sore.
        If  my back gets sore,  I take a hot bath 

Therefore:  If I work in the garden then I take a hot bath.

9.  Translate the following argument into symbolic form.  Then use a truth table to determine if the argument is valid. 

        If Emilio and Rodrigo both cook, then the meal is tasty.
        Emilio cooked and the meal was not tasty.

Therefore: Rodrigo did not cook.

The argument is valid.    The premises are   pqr    and    p~r   and the conclusion is  ~q.
The last column of the required truth table is all T.

10. a.  What is a contradiction in logic?  Give an example of one. 

A compound logical statement that is always false.  e.g.   p~p

 
    b. We discussed the fact that one could use an invalid argument but reach a true conclusion. Put in 
         premise 2  that shows this can happen, and also is an example of a common logical fallacy.

                 Premise 1.           If Florida is a state then Maine is a state.        
                  
                 Premise 2            Maine is a state.

                  Conclusion:       Florida is a state.

11.  Use Euler diagrams to determine whether each of the following arguments is valid:

     a.         All colleges have deans
                 Wells has a dean
           Therefore: Wells is a college	Invalid.	   (I will do the diagram in class for problem 11.)	     		

     b.     All teachers occasionally make mistakes.
	 Perfect people never make mistakes.
           Therefore:  No teachers are perfect people.	

                  Valid.

12.  The truth table below defines a new logic symbol #.   

  p 	  q	 p#q	  (p#q)p    	
				
  T	  T	   F	T	
  T	  F	   T	T	
  F	  T	   T	F	
  F	  F	   F	T	

a. Fill in the empty column. 

*b.  Write p#q as simply as possible without using the # symbol.  You may use any of the symbols we 
         studied.  
                                   ~pq

13. A pirate leaves a note in the cupboard of a house describing where there is hidden treasure.  If all the statements below are true, use logic to find the location of the treasure.  Explain your reasoning.  

      a.  If this house is next to a lake, then the treasure is not in the kitchen.
      b.  If the tree in the front yard is an elm, then the treasure is in the kitchen.
      c. The house is next to a lake.
      d.  The tree in the front yard is an elm or the treasure is buried under the flagpole.
      e.  If the tree in the back yard is an oak, then the treasure is in the garage.

                       Under the flagpole.  (I’ll do it in class.)