7.4 Rules of Replacement II Trans, Impl, Equiv, Exp, Taut.

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7.4 Rules of Replacement II Trans, Impl, Equiv, Exp, Taut

14. Transposition (Trans) (P  Q) :: (~Q  ~P) If Renée is a Californian then she is from the west coast. If she is not from the west coast, then she is not a Californian. P  Q::~Q  ~P TTTTFTF TFFTTFF FTTTFTT FTFTTTT

14. Trans 1. ~(T v R)  S 2. ~S  (T v R)1, trans 1. H v (K  ~D) 2. H v (D  ~K)1, trans

14. Trans Trans can be used to set up HS. 1. A  B 2. ~C  ~B 3. B  C2, Trans 4. A  C1,3 HS

15. Impl 15. Material Implication (Impl) (P  Q) :: (~P v Q) “If you bother me then I will punch you in the nose.” “Either you stop bothering me or I will punch you in the nose.”

15. Impl Impl can be used to set up HS. 1. ~A v B 2. ~B v C 3. A  B1, Impl 4. B  C2, Impl 5. A  C3,4 HS

15. Impl 1. (G  R)  (H v B) 2. G v ~H/ R v B 3. (G  R)  (~H  B) 1, Impl 4. R v B1,3, CD

16. Equiv 16. Material Equivalence (Equiv) (P  Q) :: [(P  Q)  (Q  P)] “P iff Q” :: “if P then Q, and if Q then P” (P  Q) :: [(P  Q) v (~Q  ~P)] “P iff Q” :: “P and Q are both true, or they are both false”

17. Exp 17. Exportation (Exp) [(P  Q)  R] :: [P  (Q  R)] “If we have P, then if we have Q we have R” “If we have both P and Q, then we have R”

17. Exp Exportation can be used to set up MT. 1. A  (B  C) 2. ~C 3. (A  B)  C1, Exp 4. ~(A  B) 2,3 MT

18. Taut 18. Tautology (Taut) P :: P v PP :: P  P

14. Trans(P  Q) :: (~Q  ~P) 15. Impl(P  Q) :: (~P v Q) 16. Equiv(P  Q) :: [(P  Q)  (Q  P)] 17. Exp[(P  Q)  R] :: [P  (Q  R)] 18. TautP :: P v P P :: P  P

7.4.1 p. 52 Provide logically equivalent statements using the rules of replacement. Do Do the evens if you were born on an even numbered day. Do odds if you were born on an odd numbered day.

1. S

S v S Taut S  STaut ~~SDN

2. A  (F  B)

~~A  (F  B)DN A  (B  F)Com [A  (F  B)]  [(F  B)  A] Equiv (A v A)  (F  B) Taut

3. B v (H v J)

(H v J) v B Com B v (J v H)Com (B v H) v JAssoc

4. J  (K  P)

~J v (K  P) Impl ~(K  P)  ~J Trans J  [(K  P)  (P  K)]Equiv J  [(K  P) v (~P  ~K)] Equiv

5. J  (K  P)

6. [F  (V  F)]  [(Z  G) v (G  T)]

7. R  [(W  V)  Q]

8. S v (P v A)

9. ~(A  G)

10. ~(A  G) v ~P

(J  R)  H (R  H)  M ~(P v ~J)/ M  ~P ___________1, Exp ___________2, 4, HS ___________3, DM ___________6, DN ___________7, Simp ___________7, Com ___________9, Simp ___________5, 10, MP ___________8, 11, Conj