-- LostmyZ? - 04 Dec 2004 Exercise 8.9 b through d Problem (b)

\[\exists x \exists y(Pxu \land Pyv)\frac{v}{u}\frac{fuv}{v}=\exists x\exists y(Pxv \land Pyfuv)\]
Solution:
\[=\exists x\exists y(Pxu\frac{v}{u}\frac{fuv}{v}\land Pyv\frac{v}{u}\frac{fuv}{v})\]
\[=\exists x\exists y(Pxv \land Pyfuv)\]
Problem (c)
\[\exists x\exists y(Pxu \land Pyv)\frac{u}{x}\frac{x}{u}\frac{fuv}{v}=\exists w\exists y(Pwx \land Pyfuv)\]
Solution:
\[=\exists x\exists y(Pxu \land Pyv)\frac{x}{u}\frac{fuv}{v}\frac{w}{x}\]
\[=\exists \fbox{$x$}\exists y(Pxu\frac{x}{u}\frac{fuv}{v}\frac{w}{x} \land Pyv\frac{x}{u}\frac{fuv}{v}\frac{w}{x})\]
\[=\exists w\exists y(Pwx \land Pyfuv)\]
Problem (d)
\[[\forall x\exists y(Pxy \land Pxu) \lor\exists ufuu\equiv x]\frac{x}{x}\frac{fxy}{u}=\forall v\exists w(Pvw \land Pvfxy) \lor\exists ufuu\equiv x\]
Solution:
\[=[\forall x\exists y(Pxy \land Pxu)\frac{fxy}{u}\frac{v}{x}\frac{w}{y}\lor\exists ufuu\equiv x]\]
\[=\forall v\exists w(Pvw\frac{fxy}{u}\frac{v}{x}\frac{w}{y} \land Pvu\frac{fxy}{u}\frac{v}{x}\frac{w}{y}) \lor\exists ufuu\equiv x\]
\[=\forall v\exists w(Pvw \land Pvfxy) \lor\exists ufuu\equiv x\]

These look correct to me, except that the boxed $x$ on the second line of the solution to (c) should have been a $w$. -- ShaughanLavine - 06 Dec 2004