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
I your OP you said:
There is nothing in the directions about putting these into symbolic form. So I ask you if you are required to do so? If not why do it?
It seems to me the directions “to negate the following statements and then express again in English” are very straightforward. So once again, I do not follow your symbolic translations. So about the symbols I cannot say. Sorry.
Oh I am sorry Plato. I probably didn't say it correctly. Yes , I am required to put this in the form of symbolic equations, complete the negation using
various rules about the quantifiers and then translate back into the English. The book "How to Prove it" is teaching in chapter 2 , the use of
quantifiers. Author did give some examples and I am doing the exercises....
Yes Plato, I see the point. I think author just wants the students to get the hang of it I guess. But the way you have written the affirmative part of part a) looks different from the way I have done. Or I am doing the same thing ?
Probably the mathematical statements , when negated , won't be so hard to interpret to us mortals......
Thanks. So I am correct but did differently. These quantifiers are really brain twisters I guess. But understanding it is of great importance to appreciate
the logical structure in mathematics. One of the books which I got is Copis's Introduction to Logic. Apart from symbolic logic, he goes in great detail about
discussing general logic , which is applied in various fields. So is mathematical logic a subset of philosophical/general logic ? Because, Copi discusses lot of
things ( usually having some Latin/Greek names) without any symbols.
First, I pointed out in another thread that it's easier to check your work if you choose mnemonic names for predicates.
The closing parenthesis after P(x) should be moved to the end of the formula.The statement would be
Correct. I prefer to turn the disjunction into an implication , so the English version is "Someone is majoring in mathematics and none of his friends needs help with homework," as Plato said. It is more natural to say "All people are mortal" (using an implication) than "Everyone is either not a person or is mortal" (using a disjunction).
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.
The negated formula is correct; the English translation is not. The formula corresponding to the English phrase isb)Everyone has a roommate who dislikes everyone.
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.
Again, I prefer to turn into . Then the English translation is "There is a person such that all of his/her roommates love somebody."
Or: Everyone in the freshman class has a roommate.c)There is someone in the freshman class who doesn't
have a roommate...
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.
I would say the direct formalization isd)Everyone likes someone,but no one likes everyone.
d)let M(x,y)= x likes y
The statement would be
This is equivalent to your formula, but it is closer to English because the main connective is a conjunction, which corresponds to "but."
The negated formula is correct. Concerning English, I think "there is someone who doesn't like everyone" means . To express , I would say, "there is someone who doesn't like anybody."The negated statement would be
Translation:
Either there is someone who likes everyone or there is someone who doesn't like
everyone.