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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. [.$\lor$
  6. [.$\imp$
  7. [.$P$ $x$ $y$ ]
  8. [.$Q$ $x$ $y$ ]
  9. ]
  10. [.$\imp$
  11. [.$P$ $y$ $x$ ]
  12. [.$R$ $x$ $y$ ]
  13. ]
  14. ]
  15. ]
  16. ]
  17. \end{tikzpicture}
  18. \begin{itemize}
  19. \item $x=a$
  20. \end{itemize}
  21. \begin{multicols}{2}
  22. $$x=a \land y=a$$
  23. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  24. \Tree
  25. [.$\lor$
  26. [.$\imp$
  27. [.$P$ $a$ $a$ ]
  28. [.$Q$ $a$ $a$ ]
  29. ]
  30. [.$\imp$
  31. [.$P$ $a$ $a$ ]
  32. [.$R$ $a$ $a$ ]
  33. ]
  34. ]
  35. \end{tikzpicture}
  36. \begin{align*}
  37. &(P(a,a) \imp Q(a,a)) \lor (P(a,a) \imp R(a,a)) = \\
  38. &(\false \imp \true) \lor (\false \imp \true) = \true
  39. \end{align*}
  40. \columnbreak
  41. $$x=a \land y=b$$
  42. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  43. \Tree
  44. [.$\lor$
  45. [.$\imp$
  46. [.$P$ $a$ $b$ ]
  47. [.$Q$ $a$ $b$ ]
  48. ]
  49. [.$\imp$
  50. [.$P$ $b$ $a$ ]
  51. [.$R$ $a$ $b$ ]
  52. ]
  53. ]
  54. \end{tikzpicture}
  55. \begin{align*}
  56. &(P(a,b) \imp Q(a,b)) \lor (P(b,a) \imp R(b,a)) = \\
  57. &(\true \imp \false) \lor (\true \imp \false) = \false
  58. \end{align*}
  59. \end{multicols}
  60. \begin{itemize}
  61. \item $x=b$
  62. \end{itemize}
  63. \begin{centering}
  64. $$x=b \land y=a$$
  65. \begin{tikzpicture}[every tree node/.style={draw,circle},sibling distance=.15cm]
  66. \Tree
  67. [.$\lor$
  68. [.$\imp$
  69. [.$P$ $b$ $a$ ]
  70. [.$Q$ $b$ $a$ ]
  71. ]
  72. [.$\imp$
  73. [.$P$ $a$ $b$ ]
  74. [.$R$ $b$ $a$ ]
  75. ]
  76. ]
  77. \end{tikzpicture}
  78. \begin{align*}
  79. &(P(b,a) \imp Q(b,a)) \lor (P(a,b) \imp R(b,a)) = \\
  80. &(\true \imp \false) \lor (\true \imp \false) = \false
  81. \end{align*}
  82. \end{centering}
  83. We do not need to evaluate $x=b \land y=b$. $M\nmodels\varphi$.