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  1. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  2. \Tree
  3. [.$\exists x$
  4. [.$\forall y$
  5. [.$\imp$
  6. [.$P$ $x$ $y$ ]
  7. [.$\lor$
  8. [.$Q$ $x$ $y$ ]
  9. [.$R$ $x$ $y$ ]
  10. ]
  11. ]
  12. ]
  13. ]
  14. \end{tikzpicture}
  15. \begin{multicols}{2}
  16. $$x=a \land y=a$$
  17. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  18. \Tree
  19. % [.$\exists x$
  20. % [.$\forall y$
  21. [.$\imp$
  22. [.$P$ $a$ $a$ ]
  23. [.$\lor$
  24. [.$Q$ $a$ $a$ ]
  25. [.$R$ $a$ $a$ ]
  26. ]
  27. ]
  28. % ]
  29. % ]
  30. \end{tikzpicture}
  31. \begin{align*}
  32. &P(a,a) \imp (Q(a,a) \lor R(a,a)) = \\
  33. &\true \imp (\true \lor \true) = \true
  34. \end{align*}
  35. \columnbreak
  36. $$x=a \land y=b$$
  37. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  38. \Tree
  39. % [.$\exists x$
  40. % [.$\forall y$
  41. [.$\imp$
  42. [.$P$ $a$ $b$ ]
  43. [.$\lor$
  44. [.$Q$ $a$ $b$ ]
  45. [.$R$ $a$ $b$ ]
  46. ]
  47. ]
  48. % ]
  49. % ]
  50. \end{tikzpicture}
  51. \begin{align*}
  52. &P(a,b) \imp (Q(a,b) \lor R(a,b)) = \\
  53. &\true \imp (\true \lor \false) = \true
  54. \end{align*}
  55. \end{multicols}
  56. \begin{centering}
  57. $$x=a \land y=c$$
  58. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  59. \Tree
  60. % [.$\exists x$
  61. % [.$\forall y$
  62. [.$\imp$
  63. [.$P$ $a$ $c$ ]
  64. [.$\lor$
  65. [.$Q$ $a$ $c$ ]
  66. [.$R$ $a$ $c$ ]
  67. ]
  68. ]
  69. % ]
  70. % ]
  71. \end{tikzpicture}
  72. \begin{align*}
  73. &P(a,c) \imp (Q(a,c) \lor R(a,c)) = \\
  74. &\false \imp (\true \lor \true) = \true
  75. \end{align*}
  76. \end{centering}
  77. We have found an $x$, such that for all $y$ the formula evaluates to \textit{true}. Therefore $$M \models \varphi.$$