ITEC 122 2008fall ibarland

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exam01-practice
exam 1 practice

Instructions: You can have one single-side 8.5″x11″ piece of paper which has nothing but your handwriting/drawing on it. (No mechanical reproduction.)

This is longer than the in-class exam would be (this practice-exam should take ≤90min.)

1. (Rosen §1.1, p.17, #7): English → Propositional Logic, re freezing and snowing.
2. (Rosen §1.1, p.17, #9): English → Propositional Logic, re speeding and tickets
3. Consider the statement “If I eat sushi, I am happy.”.
   1. What is the converse of this statement?
   2. What is the contrapositive of this statement?
   3. Which of the above two is/are equivalent to the original?
4. (Rosen §1.2, p.28, #9). Tautology via truth table
5. (Rosen p.28, #11a). Use propositional equivalences to show that the following is a tautology (that is, equivalent to True): (p∧q)→p.

   Your answer will be a series of formulas, joined by ≡, with each line justified by one of the reasons from Rosen's tables 6,7,8 in section 1.2. (Fifth edition: tables 5,6,7.) You may combine any associativity or commutativity steps with other steps; just cite that additional justification.

   
   (See solution.)
6. (Rosen §1.3, p.46, #3). evaluating predicates (states and capitals)
7. (Rosen §1.3, p.47, #7). first-order logic → English: funny comedians
8. (Rosen §1.3, p.47, #9). English → first-order: Russian C++
9. (Rosen §1.3, p.47, #11). Evaluating first-order formulas: x=x²?
10. (Rosen §1.3, p.47, #21). Making interpretations: old people studying discrete math
11. (Rosen §1.4 p.59, #11). English → Nested quantifiers: students and profs asking questions
12. (Rosen §1.6, p.84, #3). Direct proof of ∀n.even(n)→even(n²).
13. Fill in the blank below with a logic formula formally defining the set-difference: For sets A and B,
    A-B = { x |                                                                                  } (See solution; the answer is also given inside the chapter.)
14. (Rosen §2.2, p.130, #15a). Proving the set version of DeMorgan's law, showing each a subset of each other.
15. (Rosen p.131, #33). Practice internalizing a definition: symmetric difference.
16. Consider the standard programming function parseDouble (abbreviation: “pd”) which takes in any String and either returns what double the string represents (in base 10), or Double.NaN (a special sentinel value, used if the string doesn't represent any double)¹ For example, pd("+18")=18.0, pd("001.9")=1.9, pd("1.0000000000000000001")=1.0 (note the round-off error inherent to type double), and pd("w00t")=Double.NaN,
    • Is this function onto (for the stated domain and codomain)? Explain.
    • Is this function one-to-one (for the stated domain and codomain)? Explain.

• (Rosen §2.2, p.130, #29). Practice with ∪,∩-: making conclusions.
• (Rosen §2.2, p.130, #31). An if-and-only-if proof: the set version the contrapositive.

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©2008, Ian Barland, Radford University Last modified 2008.Oct.16 (Thu)

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