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

So the negated statement would be

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

so the nagated statement would be

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

So the negated statement is

Translation:

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

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

The statement would be

The negated statement would be

Translation:

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

everyone.

Please comment

Thanks