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papa - Math 210-011 Final Exam Spring 2011 NAME There are...

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Math 210-011 Final Exam, Spring 2011 NAME: There are 12 questions on 12 pages . The points per page are 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6. [2 points] Question 1a. Construct a truth table for P ( P Q ). [2 points] Question 1b. Construct a truth table for ( P Q ) P ( ¬ Q ) . [1 point] Question 1c. Are either 1a or 1b tautologies? 1
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Question 2. Classify each of the following as true or false. If true, give a general proof. If false, explicitly give a specific counterexample. [2 points] (i) True or false: ( A \ B ) C = ( A C ) \ B . [3 points] (ii) True or false: A \ ( B C ) = ( A \ B ) ( A \ C ). 2
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[5 points] Question 3. Let S = { 1 , 2 , 3 } , and let R be the following relation defined on S : R = (1 , 1) , (1 , 2) , (3 , 2) , (3 , 3) , (2 , 3) , (2 , 1) Is R an equivalence relation? Why or why not? 3
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Question 4. Let A = { 1 , 2 , 3 , 4 , 5 } and let B = { 2 , 3 , 5 , 7 , 11 } . Define functions f : A A and g : A B by f = (1 , 3) , (2 , 1) , (3 , 4) , (4 , 2) , (5 , 5) g = (1 , 2) , (2 , 3) , (3 , 5) , (4 , 7) , (5 , 11) [3 points] (i) Which of f f , f g , g f are defined? Write out the ones that are defined.
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