CMSC 250 KEY Quiz #4 Monday, Oct. 1, 2001

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the quiz is 30, and you will have exactly 15 minutes to complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.
  1. [10 pnts.] Use an Euler diagram to determine if each of the following represents a valid argument. Make sure to label the parts of the diagram.

      All cats are nice.
      Some cats are yellow.
      -----
      therefore: Some yellow things are nice.
       
      Circle One: (Valid) Invalid
       
       
       
       
       
       
       
       
       
       

      Some people are frightened.
      All frightened people act strange.
      -----
      therefore: All people act strange.
       
      Circle One: Valid (Invalid)
       
       
       
       
       
       
       
       
       
       

    $\downarrow$ TURN OVER $\downarrow$

  2. [6 pnts.] Write either ``$\forall MP$'' or ``$\forall MT$'' or `` $\forall instantiation$'' to tell which rule was used to reach the conclusion shown or say that the argument is ``not valid'' by any of these.

    a) All cows eat grass.  
      Bessy my pet eats grass. NOT VALID
      therefore Bessy is a cow  
    b) No tigers eat grass.  
      Bessy my pet eats grass. $\forall MT$
      therefore Bessy is not a tiger.  
    c) All grass eating animals make good pets.  
      Bessy my pet eats grass. $\forall MP$
      therefore Bessy is a good pet.  


  3. [14 pnts.] Use the handout of the ``Logical Equivalence Rules'' and the ``Rules of Inference'' to prove the following. It is a Valid Argument - you need to prove it without using a truth table.

    P1 $\forall x [P(x) \wedge Q(x)]$
    P2 $\forall y [R(y) \rightarrow \sim Q(x)]$
    P3 $\forall z [\sim P(x) \vee M(x)]$
    P4 $P(a)$
      $ --------------$
      therefore $\exists x [M(x) \wedge \sim R(x)]$

    Line # Logical Statement Name of Rule Line Numbers Used
    1 $\sim P(a) \vee M(a)$ $\forall Instantiation$ P3
    2 $\sim \sim P(a)$ Double Neg P4
    3 $M(a)$ Disjunctive Syll. 1,2
    4 $P(a) \wedge Q(a)$ $\forall Instantiation$ P1
    5 $Q(a)$ Conjunctive Simp 4
    6 $\sim \sim Q(a)$ Double Negative 5
    7 $\sim R(a)$ $\forall MT$ P2,6
    8 $M(a) \wedge \sim R(a)$ Conjunctive Add. 3,7
    9 $\exists [M(x) \wedge \sim R(x)]$ Existential Gen. 8

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John Arras
2001-10-10