Understanding Symbolic Logic, Virginia Klenk, 5th ed., Pearson Prentice Hall, 2008, Unit 18, Ex. 2e, p. 354

Prove the logical truth.

1.     (∃y)[Fy • (x)(Gx ⊃ Hxy)] 2.     Gx 3.     Fm • (x)(Gx ⊃ Hxm)] 4.     (x)(Gx ⊃ Hxm)] • Fm 5.     (x)(Gx ⊃ Hxm) 6.     Gx ⊃ Hxm 7.     Hxm 8.     Fm 9.     Fm • Hxm 10.  (∃y)(Fy • Hxy)] 11.  Gx ⊃ (∃y)(Fy • Hxy)] 12.  (x)[Gx ⊃ (∃y)(Fy • Hxy)] 13.  (∃y)[Fy • (x)(Gx ⊃ Hxy)] ⊃ (x)[Gx ⊃ (∃y)(Fy • Hxy)]

(∃y)[Fy • (x)(Gx ⊃ Hxy)] ⊃ (x)[Gx ⊃ (∃y)(Fy • Hxy)] ACP ACP 1 EI 3 Com 4 Simp 5 UI 2,6 MP 3 Simp 7,8 Conj 9 EG 2-10 CP 11 UG 1-12 CP

 

Understanding Symbolic Logic, Virginia Klenk, 5th ed., Pearson Prentice Hall, 2008, Unit 18, Ex. 2b, p. 354

 Prove the logical truth.

1.     (∃x)(y)(z)Fxyz 2.     (y)(z)Fmyz 3.     (z)Fmyz 4.     Fmyz 5.     (∃x)Fmyz 6.     (y)(∃x)Fxyz 7.     (z)(y)(∃x)Fxyz 8.     (y)(z)(∃x)Fxyz 9.     (∃x)(y)(z)Fxyz ⊃ (y)(z)(∃x)Fxyz

/(∃x)(y)(z)Fxyz ⊃ (y)(z)(∃x)Fxyz ACP 1 EI 2 UI 3 UI 4 EG 5 UG 6 UG Equiv (Com) 1-8 CP

Understanding Symbolic Logic, Virginia Klenk, 5th ed., Pearson Prentice Hall, 2008, Unit 18, Ex. 2a, p. 354

 Prove the theorem.

1.     (x)(y)Fxy 2.     (y)Fmy 3.     Fmr 4.     (∃y)Fmy 5.     (∃x)(∃y)Fxy 6.     (x)(y)Fxy ⊃ (∃x)(∃y)Fxy

/(x)(y)Fxy ⊃ (∃x)(∃y)Fxy ACP 1 UI 2 UI 3 EG 4 EG 1-5 CP

A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed,. 7.7, 20, p. 390

 Prove the logical truth.

1.     P 2.     Q • P 3.     Q 4.     (Q • P) ⊃ Q 5.     Q ⊃ (P ⊃ Q) 6.     P ⊃ Q 7.     Q 8.     (P ⊃ Q) ⊃ Q 9.     [Q ⊃ (P ⊃ Q)] • [(P ⊃ Q) ⊃ Q] 10.  Q ≡ (P ⊃ Q) 11.  P ⊃ [Q ≡ (P ⊃ Q)]

/ P ⊃ [Q ≡ (P ⊃ Q)] ACP ACP 2 Simp 1-4 CP 4 Exp ACP 1,6 MP 6-7 CP 5,8 Conj 9 Equiv 1,10 CP

A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed,. 7.7, 18, p. 390

 Prove the logical truth.

1.     ~ [(P • Q) v R] ⊃ [(~ R v Q) ⊃ (P ⊃ Q)] 2.     ~ [~ (P • Q) v R] v [(~ R v Q) ⊃ (P ⊃ Q)] 3.     [(P • Q) v R] • ~ [(~ R v Q) ⊃ (P ⊃ Q)] 4.     [(P • Q) v R] • ~ [~ (~ R v Q) v (P ⊃ Q)] 5.     [(P • Q) v R] • [(~ R v Q) • ~ (P ⊃ Q)] 6.     [(P • Q) v R] • [(~ R v Q) • ~ (~ P v Q)] 7.     [(P • Q) v R] • (~ R v Q) • (P • ~ Q) 8.     P • ~ Q 9.     ~  Q 10.  ~ R v Q 11.  ~ R 12.  (P • Q) v R 13.  P • Q 14.  Q 15.  Q • ~ Q 16.  ~ ~ [(P • Q) v R] ⊃ [(~ R v Q) ⊃ (P ⊃ Q)] 17.  [(P • Q) v R] ⊃ [(~ R v Q) ⊃ (P ⊃ Q)]

/[(P • Q) v R] ⊃ [(~ R v Q) ⊃ (P ⊃ Q)] IP 1 Impl 2 DM 3 Impl 4 DM 5 Impl 6 DM 7 Simp 8 Simp 7 Simp 9,10 DS 7 Simp 11,12 DS 13 Simp 9,14 Conj 1-15 IP 16 DN

A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed,. 7.7, 17, p. 390

 Prove the logical truth.

1.     P 2.     Q • ~ Q 3.     Q 4.     Q v R 5.     ~ Q • Q 6.     ~ Q 7.     R 8.     (Q • ~ Q) ⊃ R 9.     P ⊃ [(Q • ~ Q) ⊃ R]

/ P ⊃ [(Q • ~ Q) ⊃ R] ACP ACP 2 Simp 3 Add 2 Com 5 Simp 4,6 DS 2-7 CP 1-8 CP

A Concise Introduction to Logic, Patrick J. Hurley, Wadsworth, 2006, 9th ed,. 7.7, 15, p. 390

 Prove the logical truth.

1.     ~ P v Q 2.     P v ~ Q 3.     P ⊃ Q 4.     ~ Q v P 5.     Q ⊃ P 6.     (P ⊃ Q) • (Q ⊃ P) 7.     P ≡ Q 8.     (P v ~ Q) ⊃ (P ≡ Q) 9.     (~ P v Q) ⊃ [(P v ~ Q) ⊃ (P ≡ Q)]

/ (~ P v Q) ⊃ [(P v ~ Q) ⊃ (P ≡ Q)] ACP ACP 1 Impl 2 Com 4 Impl 3,5 Conj 6 Equiv 2-7 CP 1-8 CP