Construct direct proofs to show that the following symbolic arguments are valid.  Commas mark the breaks between premises.  F→(G→H), ~F→J, ~(G→H) ∴ J P→Q , R→~S, P v R, (Q v ~S)→(~T v ~W), ~~T ∴ ~W  (A v G)→K,  K→(B→F),  A∙B ∴ F

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Construct direct proofs to show that the following symbolic arguments are valid.  Commas mark the breaks between premises. 

  1. F→(G→H), ~F→J, ~(G→H) ∴ J
  2. P→Q , R→~S, P v R, (Q v ~S)→(~T v ~W), ~~T ∴ ~W 
  3. (A v G)→K,  K→(B→F),  A∙B ∴ F 
  4. ~C→(F→C), ~C ∴ ~F 
  5. ~(C∙D), ~C→S, ~D→T ∴ S v T 
  6. (W→U)∙~X ∴ ~U→~W
  7. ~~T v ~R, ~(S v ~R), (T∙~S)→~Q , W→Q ∴ ~W
  8. ~(J∙L), (~J v ~L)→~M, ~E v (M v ~S) ∴ ~(S∙E)
  9. (B∨A)→C, ~B→D, ~D ∴ C 
  10. ~(O∙N), (~O→S)∙(~N→T) ∴ S v T
  11. ~M v N ∴ ~N→~M 
  12. ~B↔C, ~B ∴ C
  13. (Z v ~Y)∙(Z v W), Z→~~U, ~Y→(W→U) ∴ U
  14. ~U→~B, S→~B, ~(U∙~S), T v B ∴ T 
  15. A↔B, B→C ∴ ~A v C 
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