Exam 1 grading guidelines (and solutions)

1. Answer: Via transistors (IC chips is also acceptable)
   • We looked for certain buzzwords related to implementing logic gate, such as: Transistors, Half-adders, Integrated Circuit, IC, Electrical current flow (high/low)
   • 5 points for any of the above answers
   • 2 points if they just said “high/low” or were off a lot
   • 1 point if they attempted anything
   • 0 point for blanks
      
2. Answer: A trivial proof is when the consequence is true; a vacuous proof is when the antecedent is false;
   • 5 points if they mentioned “consequent is always true” and “antecedent is false, so consequent is true”
   • 4 points if they made a small mistake on one, (for example, “consequent is always false”)
   • 3 points if they got either one completely wrong
   • 2 points if they omitted one, and made a small mistake on the other
   • 1 point if they attempted anything
   • 0 point if blank
      
3. Answer: AI, n-queens, databases, etc.
   • 5 points if they referenced something from class or homework
   • 5 points if I could see how a declarative programming language could be applied
   • 4 points if only gave one example
   • 2 points if answer was too trivial or way off
   • 1 point if they attempted anything
   • 0 point if blank
      
4. Answer: The ability to write an algorithm that will determine if a given program will halt or loop forever when supplied with a given input.
   • 5 points if they said something like “will a program halt or not”
   • 1 point if they attempted anything
   • 0 point if blank
      
5. Answer: It was the first algorithm to be shown that it cannot ever exist (at least on a digital computer).
   • 5 points if they said something like “first program to be unsolvable”
   • 1 point if they attempted anything
   • 0 point if blank
      
6. Answer: A direct proof via the contrapositive
   • 5: Correct translation of modus tollens into English
   • 4: Minor mistake, but included some sort of example that cleared up your mistake.
   • 3: Minor mistake, answer mostly correct, e.g. used incorrect term for part of the hypothesis
   • 2: Major mistake, translated some other logical rule into English
   • 1: Something was written
      
7. Answer: If somebody is not a student, then this statement is true; it wants to show that there exists a student who has red hair; the conditional should be a conjunction.
   • The explanation of why the original statement was incorrect was worth 2 points
   • The correctly rewritten statement was worth 2 points
   • 1 point was awarded if you attempted to answer the question
      
8. Answer: A constructive existence proof shows the exact value; a non-constructive existence proof just shows that it exists, but not what the value is.
   • 5: correctly explained the difference
   • 4: mostly correct, small error, e.g. getting constructive and non-constructive reversed
   • 3: showed some knowledge of the terms, but had a major mistake, e.g. correctly explained one but not the other.
   • 2: Showed some knowledge of the terms but failed to describe either of them correctly
   • 1: There was some attempt made on the problem
      
9. Answer: Conch shell spirals, the golden ratio (which is the ratio of one's arm to one's leg), etc.
   • For the most part this was an all or nothing question
      
10. Answer: Countably infinite (the integers, rationals, ordered pairs of ints) can be listed in such a way that you will (eventually) reach each finite number; uncountably infinite (the reals) cannot be listed in such a manner
    • 5: correctly explained the difference, and gave a valid example.
    • 4: explanation showed understanding of the terms, but some small mistake was made.
    • 3: explanation showed some understanding of the terms, but failed to correctly define them, valid example given.
    • OR correct explanation with incorrect example given.
    • 2: incorrect or missing explanation, valid example
    • 1: there was some attempt made on the problem
       
11. Answer: He things they are a waste of ink, and a scam to get you to pay more money to the printer companies
    • This question was a bonus to those who were in class that day.  The answer of, "he thinks they are a waste of ink" got full credit.  Other answers were awarded points based on how likely it was the student was in class that day.
       
12. Answer: Give it a value or provide a quantifier
    • 5 points for both of the correct answers
    • 3 points for just one correct answer
       
13. Answer: yes or no
    • All or nothing, as long as they answered this yes or no
       
14. Answer: An indirect proof is a direct proof via the contrapositive
    • 5 points for getting this all right
    • 3 points for having the right idea, but not quite there
    • 1 point for writing something down, even if it's totally wrong
       
15. Answer: see the textbook (page 61) for the diagram
    • 2 Points for attempting the problem.
    • 4 Points for correctly computing the carry.
    • 4 Points for correctly computing the sum.
    • Total: 10 Points
       
16. Answer: ...
    • Note that the '-' should be a '+'
    • If the proof wasn't fully correct, then points were awarded based on how "good" the proof was
       
17. Answer: ...
    • 2 Points for attempting the problem.
    • 4 Points for showing an understanding of propositional logic and how to use the given laws.
    • 4 Points for finishing the proof correctly.