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CMSC 250 Quiz #3 ANSWERS Wed., Sept. 15, 2004

  1. [12 pnts.] Prove that the following argument is valid by using the rules. Apply only the rules from Theorem 1.1.1 and Table 1.3.1 and the other things listed on the ``cheat sheet''. You must tell the name of the rule as you apply it and what line numbers contained the statements used to apply the rule.
    P1 $q \vee \sim p$
    P2 $r \rightarrow \sim q$
    P3 $p \rightarrow r$
      therefore $ \sim p$
    # Statement Rule Lines Used
    1 $\sim p \vee q$ Commutative P1
    2 $p \rightarrow q$ Def of Impl. 1
    3 $\sim r \vee \sim q$ Def of Impl. P2
    4 $\sim q \vee \sim r$ Commutative 3
    5 $q \rightarrow \sim r$ Def of Impl. 4
    6 $p \rightarrow \sim r$ Hypothetical Syll. 2,5
    7 $\sim p \vee \sim r$ Def of Impl. 6
    8 $\sim p \vee r$ Def of Impl. P3
    9 $(\sim p \vee \sim r) \wedge (\sim p \vee r)$ Conjunctive Add. 7,8
    10 $\sim p \vee (\sim r \wedge r)$ Distributive 9
    11 $\sim p \vee c$ Negation Law 10
    12 $ \sim p$ Identity 11
    or
    # Statement Rule Lines Used
    1 $\vert p$ Assume  
    2 $\vert r$ Modus Ponens 1,P3
    3 $\vert\sim q$ Modus Ponens 2,P2
    4 $\vert\sim q \wedge p$ Conjunctive Add 1,3
    5 $\vert\sim (q \vee \sim p)$ DeMorgan's 4
    6 $ \sim p$ Closing Cond Wrld w/ Contradiction 1-5
          p1,5




    $\downarrow$ TURN OVER $\downarrow$
  2. [12 pnts.] Prove that the following argument is valid by using the rules. Apply only the rules from Theorem 1.1.1 and Table 1.3.1 and the other things listed on the ``cheat sheet''. You must tell the name of the rule as you apply it and what line numbers contained the statements used to apply the rule.
    P1 $p \vee r$
    P2 $\sim (q \wedge \sim r)$
      therefore $(p \rightarrow q) \rightarrow r$
    # Statement Rule Lines Used
    1 $\sim q \vee \sim \sim r$ DeMorgan's P2
    2 $\sim q \vee r$ Double Neg 1
    3 $(p \vee r) \wedge (\sim q \vee r)$ Conjunctive Add P1,2
    4 $(p \wedge \sim q) \vee r$ Distributive 3
    5 $\sim (p \wedge \sim q) \rightarrow r$ Def of Impl. 4
    6 $(\sim p \vee \sim \sim q) \rightarrow r$ DeMorgan's 5
    7 $(\sim p \vee q) \rightarrow r$ Double Neg 6
    8 $(p \rightarrow q) \rightarrow r$ Def of Impl. 7
    or
    # Statement Rule Lines Used
    1 $\sim q \vee \sim \sim r$ DeMorgan's P2
    2 $\sim q \vee r$ Double Neg 1
    3 $\sim p \rightarrow r$ Def of Impl. P1
    4 $q \rightarrow r$ Def of Impl. 2
    5 $\vert p \rightarrow q$ Assume  
    6 $\vert \sim p \vee q$ Def of Impl. 5
    7 $\vert r$ Dilemma 3,4,6
    8 $(p \rightarrow q) \rightarrow r$ Close of Conditional Wrld 5-7
  3. [6 pnts.] For each of the following identify the rule being applied or tell that it is not one of those listed on the ``cheat sheet''.
    1. If I stay up late at night, then I wil be tired in the morning.  
      I am not tired this morning. MODUS TOLLENS
      Therefore I did not stay up late last night.  




    2. Either I study or I like football.  
      If I like football, then I like mathematics. NONE
      If I don't study, then I like mathematics.  




    3. If I work hard, then I earn lots of money.  
      If I earn lots of money, then I pay high taxes. NONE
      If I pay high taxes, then I have worked hard.  

Kin-Keung Ma 2004-09-15

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