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Provide an expression using predicates that describes the sentence.

INSTRUCTIONS TO CANDIDATES
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Questions

Part 1: Propositional logic and predicates

1. (4 points, anticipated 5 min. length) Use predicates to explain the concept that, you cannot drive on any road and you cannot arrive at any location when there is no gas in the car.

1. Define the variables described in the statement. For example:

Let such and such be that such and such...

2. Provide an expression using predicates that describes the sentence.

2. (Anticipated 20 min. length) Let t, u and v be propositional random variables. Use a truth table to determine the truth of the following statements:

1. (4 points) tɅ (u ^ ¬v)

2. (4 points) (t v u) vv → (¬u → t)

3. (8 points, anticipated 10 min. length) Use a direct proof to prove that the product of two even numbers is even. Number each line of the proof, and note each major step. Basic algebra, such as multiplication, does not need to be annotated.

4. (Anticipated 20 min. length) This question pertains to counting infinite sequences.

1. (4 points) Describe the condition(s) needed to determine if two sets are the same size. Provide 3-5 sentences explaining all necessary concepts.

2. (4 points) Provide work attempting to count the sequence R*, the set of positive real numbers, restricted to [0,1].

3. (2 points) Following the previous problem, if a notation can be given to describe the set's size, provide this notation. Unlike other problems it is OK to be brief for this sub-problem.

5. (Anticipated 25 min. length) Let p be that you ate a peanut, let q be that you are sneezing and let r be that your eyes water. For each of the following statements, write out both the propositional logic expression described by the statement and it's English version. Write out the statements explicitly, however nonsensical they may seem, without simplying them.

1. (2 points) Write out this statement: p → q.

2. (2 points) Write out this statement: q v r.

3. (2 points) Write out this statement: p^q→r.

4. (4 points) If you ate a peanut then you are sneezing. We know that you ate a peanut. We can infer that you are sneezing. Additionally, what is this called, and is it true?

 

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