Section 2.6: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Find prenex formulas equivalent to the following.

(a) $(\exists xAx\wedge\exists xBx)\rightarrow Cx$ .

(b) $\forall xAx\leftrightarrow\exists xBx$ .

(a) The following transformations are based on tautological equivalence, and the EAV and Prenex Normal Form theorems.

$(\exists xAx\wedge\exists xBx)\rightarrow Cx$

$\exists xAx\rightarrow\exists xBx\rightarrow Cx$

$\exists zAz\rightarrow\exists yBy\rightarrow Cx$

$\exists zAz\rightarrow\forall y(By\rightarrow Cx)$

$\forall z(Az\rightarrow\forall y(By\rightarrow Cx))$

$\forall z\forall y(Az\rightarrow By\rightarrow Cx)$

$\forall z\forall y((Az\wedge By)\rightarrow Cx)$

(b) The following transformations are based on tautological equivalence, and the EAV and Prenex Normal Form theorems.

$\forall xAx\leftrightarrow\exists xBx$

$(\forall xAx\rightarrow\exists yBy)\wedge(\exists zBz\rightarrow\forall wAw)$

$\neg((\forall xAx\rightarrow\exists yBy)\rightarrow\neg(\exists zBz\rightarrow\forall wAw))$

$\neg(\exists x(Ax\rightarrow\exists yBy)\rightarrow\neg\forall z(Bz\rightarrow\forall wAw))$

$\neg(\exists y\exists x(Ax\rightarrow By)\rightarrow\neg\forall w\forall z(Bz\rightarrow Aw))$

$\neg(\exists y\exists x(Ax\rightarrow By)\rightarrow\exists w\exists z\neg(Bz\rightarrow Aw))$

$\neg\exists z\exists w(\exists y\exists x(Ax\rightarrow By)\rightarrow\neg(Bz\rightarrow Aw))$

$\neg\forall x\forall y\exists z\exists w((Ax\rightarrow By)\rightarrow\neg(Bz\rightarrow Aw))$

$\exists x\exists y\forall z\forall w\neg((Ax\rightarrow By)\rightarrow\neg(Bz\rightarrow Aw))$

$\exists x\exists y\forall z\forall w((Ax\rightarrow By)\wedge(Bz\rightarrow Aw))$

Note. In both cases it might be easier to get the answer by considering when the sentence is false. (a) is false when the following is true, $\exists xAx\wedge\exists xBx\wedge\neg Cx$ , $\exists zAz\wedge\exists yBy\wedge\neg Cx$ , $\exists z\exists y(Az\wedge By\wedge\neg Cx)$ , $\exists z\exists y\neg((Az\wedge By)\rightarrow Cx)$ , $\neg\forall z\forall y((Az\wedge By)\rightarrow Cx)$ . (b) is false when the following is true, $\forall xAx\wedge\forall x\neg Bx$ or $\exists x\neg Ax\wedge\exists xBx$ , $\forall xAx\wedge\forall y\neg By$ or $\exists w\neg Aw\wedge\exists zBz$ , $\forall x\forall y(Ax\wedge\neg By)$ or $\exists z\exists w(\neg Aw\wedge Bz)$ , $\forall x\forall y\neg(Ax\rightarrow By)$ or $\exists z\exists w\neg(Bz\rightarrow Aw)$ , $\neg\exists x\exists y(Ax\rightarrow By)$ or $\neg\forall z\forall w(Bz\rightarrow Aw)$ .

Subpages

Section 2.6: Problem 5 Solution

Section 2.6: Problem 5 Solution