Symbolization key:
- $E(x)$: x is even.
- $O(x)$: x is odd
$ \def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\} \def\Ai#1{\qquad\mathbf{\forall I} \: #1 \\} \def\Ee#1{\qquad\mathbf{\exists E} \: #1 \\} \def\Ei#1{\qquad\mathbf{\exists I} \: #1 \\} \def\R#1{\qquad\mathbf{R} \: #1 \\} \def\ii#1{\qquad\mathbf{\to I} \: #1 \\} \def\ie#1{\qquad\mathbf{\to E} \: #1 \\} \def\be#1{\qquad\mathbf{\leftrightarrow E} \: #1 \\} \def\bi#1{\qquad\mathbf{\leftrightarrow I} \: #1 \\} \def\qi#1{\qquad\mathbf{=I}\\} \def\qe#1{\qquad\mathbf{=E} \: #1 \\} \def\ne#1{\qquad\mathbf{\neg E} \: #1 \\} \def\IP#1{\qquad\mathbf{IP} \: #1 \\} \def\x#1{\qquad\mathbf{X} \: #1 \\} $ $ \fitch{1.\, \forall x(E(x) \leftrightarrow \exists y(x = 2y)) \\2.\, \forall x(O(x) \leftrightarrow \exists y(x = 2y + 1)) }{ \fitch{3.\, E(a)}{ 4.\, E(a) \leftrightarrow \exists y(a = 2y) \Ae{1} 5.\, \exists y(a = 2y) \be{4,3} \fitch{6.\, a = 2k}{ 7.\, a + 1 = a + 1 \qi{} 8.\, a + 1 = 2k + 1 \qe{6,7} 9.\, \exists y(a + 1 = 2y + 1) \Ei{8} 10.\, O(a + 1) \leftrightarrow \exists y(a + 1 = 2y + 1) \Ae{1} 11.\, O(a + 1) \be{10,9} }\\ 12.\, O(a + 1) \Ee{5,6-11} }\\ \fitch{13.\, O(a+1)}{ 14.\, O(a+1) \leftrightarrow \exists y(a + 1 = 2y + 1) \Ae{2} 15.\, \exists y(a + 1 = 2y + 1) \be{13,14} \fitch{16.\, a + 1 = 2k + 1}{ 17.\, a = 2k \qquad \text{aritmethic}\\ 18.\, \exists y(a = 2y) \Ei{17} 19.\, E(a) \leftrightarrow \exists y(a = 2y) \Ae{1} 20.\, E(a) \be{19,18} }\\ 21.\, E(a) \Ee{15,16-20} }\\ 22.\, E(a) \leftrightarrow O(a+1) \bi{3-12,13-21} 23.\, \forall x(E(x) \leftrightarrow O(x+1)) \Ai{22} } $
Is the conclusion correctly symbolized using quantifiers ? Is this proof correct ?