Question

The negation of the following two quantified predicates.

$\forall x\forall y\left[\right(x<y)\to \exists z(x<z<y\left)\right]$ is

Which of the following statement is true?

• A. $\exists x\exists y\exists z\left[\right(z\le x<y)\wedge (x<y\le z\left)\right]$
• B. $\exists x\exists y\exists z\left[\right(z\le x<y)\vee (x<y\le z\left)\right]$
• C. $\exists x\exists y\exists z\left[\right(z\le x<y)\leftrightarrow (x<y\le z\left)\right]$
• D. $\exists x\exists y\exists z\left[\right(z\le x<y)\to (x<y\le z\left)\right]$

Answer 1

The correct option is B $\exists x\exists y\exists z\left[\right(z\le x<y)\vee (x<y\le z\left)\right]$

option (a)

$\sim \left\{\forall x\forall y\right[(x<y)\to \exists z(x<z<y)\left]\right\}$

$\iff \exists x\exists y\left[\right(x<y)\wedge \forall z\sim (x<z<y\left)\right]$

$\iff \exists x\exists y\left[\right(x<y)\wedge \forall z((z\le x)\vee (y\le z)\left)\right]$

$\iff \exists x\exists y\forall z\left[\right(x<y)\wedge z((z\le x)\vee (y\le z)\left)\right]$

$\iff \exists x\exists y\forall z\left[\right((x<y)\wedge (z\le x))\wedge ((x<y)\wedge (y\le z)\left)\right]$

$\iff \exists x\exists y\exists z\left[\right(z\le x<y)\vee (x<y\le z\left)\right]$