Need help in negation of statements

[issacnewton]

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

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

So the negated statement would be

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

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

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

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

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

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

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

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

so the nagated statement would be

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

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

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

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

So the negated statement is

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

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

Translation:

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

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

The statement would be

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

The negated statement would be

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

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

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

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

Translation:

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



Please comment

Thanks

 

[issacnewton]

any help ?