Homework assignments

Homework assignment 02

1.1   Statements, Symbolic Representation, and Tautologies

1. Using letters for the component statements, translate the following compound statements into symbolic notation.
   1. If the horse is fresh, then the knight will win.

      A:	horse is fresh
      B:	knight will win
      	A -> B

   2. The knight will win only if the horse is fresh and the armor is strong.

      A:	horse is fresh
      B:	armor is strong
      C:	knight will win
      	C -> (A /\ B)

   3. A fresh horse is a necessary condition for the knight to win.

      A:	horse is fresh
      B:	knight will win
      	B -> A

   4. The knight will win if and only if the armor is strong.

      A:	armor is strong
      B:	knight will win
      	A <-> B

   5. A sufficient condition for the knight to win is that the armor is strong or the horse is fresh.

      A:	armor is strong
      B:	horse is fresh
      C:	knight will win
      	(A \/ B) -> C

2. Construct truth tables for the following wffs. Note any tautologies or contradictions.
   1. (A -> B)<-> A' \/ B

      A	B	A -> B	A' \/ B	(A -> B)<-> A' \/ B
      T	T	T	T	T
      T	F	F	F	T
      F	T	T	T	T
      F	F	T	T	T

      Tautology
3. Rewrite the following statement form with a simplified conditional expression, where the function odd(n) returns true if n is odd.

   if not((Value1 < Value2) or odd(Number))
   or (not(Value1 < Value2) and odd(Number)) then
       statement1
   else
       statement2
   end if
   

   A:	Value1 < Value2
   B:	odd(Number)

   (A \/ B)' \/ (A' /\ B)	=	(A' /\ B') \/ (A' /\ B)	(De Morgan's Laws)
   	=	A' /\ (B' \/ B)	(Distributive Property)
   	=	A' /\ 1	(Complement Property)
   	=	A'	(Identity Property)

   Thus, the statement can be rewritten:

   if Value1 >= Value2 then
       statement1
   else
       statement2
   end if
   

4. Every compound statement is equivalent to a statement using only the connectives of conjunction and negation. To see this, we need to find equivalent wffs for A \/ B and for A -> B that use only /\ and '. These new statements can replace, respectively, any occurrences of A \/ B and A -> B. (The connective <-> was defined in terms of other connectives, so we already know that it can be replaced by a statement using these other connectives.)
   1. Show that A \/ B is equivalent to (A' /\ B')'.

      A	B	A'	B'	A \/ B	(A' /\ B')'
      T	T	F	F	T	T
      T	F	F	T	T	T
      F	T	T	F	T	T
      F	F	T	T	F	F

   2. Show that A -> B is equivalent to (A /\ B')'.

      A	B	B'	A -> B	(A /\ B')'
      T	T	F	T	T
      T	F	T	F	F
      F	T	F	T	T
      F	F	T	T	T

   
   

1.2   Propositional Logic

1. Prove the following argument using propositional logic:
   (A'-> B)/\(B-> C)/\(C-> D)->(A' -> D)
   We use the deduction method to rewrite the argument with A' as further hypothesis and D as conclusion.

   1.	A'-> B	hyp
   2.	B-> C	hyp
   3.	C -> D	hyp
   4.	A'	hyp
   5.	B	mp, 1,4
   6.	C	mp, 2,5
   7.	D	mp, 3,6

2. The argument of the defense attorney at the beginning of this chapter was:

   ``If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard didn't see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent.''

   This can be written as:
   (A-> B) /\ (B' \/ C) /\ (D' -> C') /\ (D -> (B /\ E)) /\ E' -> A'

   1.	A -> B	hyp
   2.	B' \/ C	hyp
   3.	D' -> C'	hyp
   4.	D -> (B/\ E)	hyp
   5.	E'	hyp
   6.	C \/ B'	comm, 2
   7.	C' -> B'	imp, 6
   9.	B' -> A'	cont, 1
   10.	(B/\ E)' -> D'	cont, 4
   11.	(B'\/ E')	add, 5
   12.	(B/\ E)'	dm, 11
   13.	D'	mp, 10,12
   14.	C'	mp, 3,13
   15.	B'	mp, 7,14
   16.	A'	mp, 9,15

Homework assignments

Homework assignment 02