ITEC 122 2008fall ibarland

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exam01
exam 1

Instructions: Closed book, closed most-notes, but you can have one single-sided 8.5″x11″ page of notes/pictures in your own handwriting (no photocopying etc).

1. (8pts) Let n be the proposition “a null pointer exception occurs”,
   let v be the proposition “the array index is valid”, and
   let a be the proposition “the array has been initialized”.
   Give a propositional formula expressing:
   1. If a null pointer exception occurs, then the array has not been initialized.
      
                                                                  
   2. The array index is valid, but a null pointer exception occurs.
      
                                                                  
   3. Initializing the array and having a valid array index is sufficient to avoid a null pointer error.
      
                                                                  
   4. (Challenge:) Initializing the array and having a valid array index is necessary to avoid a null pointer error.
      
                                                                  
2. (6pts) Use truth tables to show that the following is a tautology: [¬q∧(p∨q)] → p.

3. (8pts) For each of the following, indicate whether or not they are true, over the domain of the integers:
   1. true or false?: ∃x. ((30 < x²) ∧ (x² < 40))
   2. true or false?: ∃x. ((30 < x²) ∨ (x² < 40))
   3. true or false?: ∀x. ((30 < x²) ∧ (x² < 40))
   4. true or false?: ∀x. ((30 < x²) ∨ (x² < 40))
4. (12pts) Let Q(x,y) be the statement “student x has been a contestant on quiz show y”. Express each of the following sentences as a first-order logic formula, where the domain for x is all RU students, and y is all TV quiz shows.
   1. “Amy McCoy IV has been a contestant on Family Feud.” (You may abbreviate Family Feud as “FF”.)
      
                                                                  
   2. “There is an RU student who has been a contestant on Jeopardy and on Family Feud”
      
                                                                  
   3. “There is a RU student who has been a contestant on a TV quiz show.”
      
                                                                  
   4. “No RU student has ever been a contestant on a TV quiz show.”
      
                                                                  
   5. “Every TV quiz show has had an RU student as a contestant.”
      
                                                                  
   6. “At least two RU students have been contestants on Jeopardy.”
      
                                                                  
5. (6pts) The symmetric difference of two sets A and B, denoted A ⊕ B, is the set of elements which are in A or in B, but not in both. Fill in the blank below with a logic formula expressing this, where that formula does not use the binary exclusive-or operator (also denoted ⊕):
   
   A⊕B = { x |                                                                                 }
6. (4pts) Draw a Venn diagram for each of the following:
   1. A-(B∩C).

   2. (A-B)∪(A-C).

7. (8pts) Prove that, for sets A,B,C,
   (A ∩ B ∩ C) ⊆ A ∪ B.
   Be sure to use the definition of ⊆, ∩, and ∪ appropriately, and any justifications you need from Rosen's Table 1 (attached).
8. (4pts) In the following, φ represents the empty set, {}, and P(A) represents the power set of A (where A is any set).
   1. true or false?: φ ∈ P({φ,{φ}})
   2. true or false?: φ ⊆ P({φ,{φ}})
   3. true or false?: {φ} ∈ P({φ,{φ}})
   4. true or false?: {φ} ⊆ P({φ,{φ}})
9. (4pts) Consider the Java method parseInt which takes in any String and returns what int the string represents (in base 10). (If the String doesn't represent a number, it does not return a value — it throws an exception.) For example, parseInt("+18")=18, and parseInt("001")=1.
   • Why is parseInt not a function from Strings to ints?

   • Concisely describe a domain and codomain for which it is a function.

10. (6pts) Use propositional equivalences to show that (a→b) ∨ a ≡ true. (that is, (a→b) ∨ a is a tautology).

    Your answer will be a series of formulas, joined by ≡, with each line justified by one of the reasons from the attached Table 6 from Rosen. You may combine any associativity or commutativity steps with other steps; just cite that additional justification.

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