Hi

I have to negate the following statements and then express again

in English. I need to know if I am making any mistakes.

a)Everyone who is majoring in math has a friend who needs

help with his homework.

b)Everyone has a roommate who dislikes everyone.

c)There is someone in the freshman class who doesn't

have a roommate.

d)Everyone likes someone,but no one likes everyone.

My answers are as follows--------

a) Let P(x)= x is majoring in math

Q(x)=x needs help with homework

M(x,y)=x and y are friends

The statement would be

$\displaystyle \exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))$

So the negated statement would be

$\displaystyle \neg \left[\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))\right]$

$\displaystyle \neg \left[\neg(\exists x (P(x)) \vee \exists y (M(x,y)\wedge Q(y))\right]$

$\displaystyle \neg \left[\forall x \neg P(x) \vee \exists y (M(x,y)\wedge Q(y))\right] $

$\displaystyle \neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y)) $

$\displaystyle \exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))$

$\displaystyle \exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))$

To translate this statement back to English would be

Some person is majoring in math and everyone is either not a

friend of this person or doesn't need help in homework.

b) let Q(x,y)=x and y are roommates

M(x,y)=x likes y

The statement would be

$\displaystyle \forall x \left[\exists y (Q(x,y) \wedge \forall z(\neg M(y,z)))\right] $

so the nagated statement would be

$\displaystyle \neg \forall x \left[\exists y (Q(x,y)\wedge \forall z(\neg M(y,z)))\right] $

$\displaystyle \exists x \forall y \left[\neg Q(x,y) \vee \exists z(M(y,z)) \right] $

Translation:

Either there is some person who is not roommate with anybody or there is

someone who is liked by all.

c)let P(x)= x is in freshman class.

M(x)=x has a roommate.

The statement would be

$\displaystyle \exists x \left[ P(x)\wedge \neg M(x) \right] $

So the negated statement is

$\displaystyle \neg \exists x \left[ P(x)\wedge \neg M(x) \right] $

$\displaystyle \forall x \left[ \neg P(x) \vee M(x) \right] $

Translation:

Everyone either is not in freshman class or has a roommate.

d)let M(x,y)= x likes y

The statement would be

$\displaystyle \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]$

The negated statement would be

$\displaystyle \neg \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]$

$\displaystyle \neg \left[\forall x \exists y (M(x,y))\wedge \forall x \exists z \neg M(x,z) \right]$

$\displaystyle (\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )$

$\displaystyle \left[\exists x \forall y \neg M(x,y)\right] \vee \left[\exists x \forall z M(x,z)\right]$

Translation:

Either there is someone who likes everyone or there is someone who doesn't like

everyone.

Please comment

Thanks