Logic problems involving quantifiers

[eku_girl83]

Here are some of my homework problems involving quantifiers. My answer is listed below the question. Could someone give me some feedback on whether my answers are correct?

1. Translate the sentence into a symbolic sentence with quantifiers.
a) Some isosceles triangle is a right triangle.
(there exists an x) (x is an isosceles triangle ^ x is a right triangle)
b) No right triangle is isosceles.
(for all x) (x is a right triangle implies x is not isosceles)
c) All people are honest or no one is honest.
(for all x) (x is honest) or (for all x) (x is not honest)
d) Some people are honest and some people are not honest.
(for all x) (x is honest or x is not honest)

2. For each proposition, write a useful denial, and translate it into ordinary English.

a) Not all precious stones are beautiful.
Denial: All precious stones are beautiful.
(for all x) (x is a precious stone implies x is beautiful)
b) No right triangle is isosceles.
Denial: There exists a right triangle which is isosceles.
(there exists x) (x is a right triangle ^ x is isosceles)
c) No one loves everybody.
Denial: someone loves everybody.
(there exists x) (x is a person ^ x loves everybody)
d) Everybody loves someone.
Denial: Everybody does not love someone.
(there exists x) (x is a person ^ x does not love someone)

I'm using the symbology of a backwards E to denote "there exists an x" and an upside down A to denote "for all x."

Can someone tell me if I'm on the right track with these?
Thanks!

[HallsofIvy]

Here are some of my homework problems involving quantifiers. My answer is listed below the question. Could someone give me some feedback on whether my answers are correct?

1. Translate the sentence into a symbolic sentence with quantifiers.
a) Some isosceles triangle is a right triangle.
(there exists an x) (x is an isosceles triangle ^ x is a right triangle)
b) No right triangle is isosceles.
(for all x) (x is a right triangle implies x is not isosceles)
c) All people are honest or no one is honest.
(for all x) (x is honest) or (for all x) (x is not honest)
d) Some people are honest and some people are not honest.
(for all x) (x is honest or x is not honest)

a, b, c look okay. d is not correct since it would be true if all people were honest. I would say "(there exist x)(x is honest) and (there exist x)(x is not honest)"

2. For each proposition, write a useful denial, and translate it into ordinary English.

a) Not all precious stones are beautiful.
Denial: All precious stones are beautiful.
(for all x) (x is a precious stone implies x is beautiful)
b) No right triangle is isosceles.
Denial: There exists a right triangle which is isosceles.
(there exists x) (x is a right triangle ^ x is isosceles)
c) No one loves everybody.
Denial: someone loves everybody.
(there exists x) (x is a person ^ x loves everybody)
d) Everybody loves someone.
Denial: Everybody does not love someone.
(there exists x) (x is a person ^ x does not love someone)

I'm using the symbology of a backwards E to denote "there exists an x" and an upside down A to denote "for all x."

Can someone tell me if I'm on the right track with these?
Thanks!

Except for 1 d, you are fine.