this is my solution of my homework, is that true 100%? Please if there is any mistake tell me because my professor is so careful.
Thanks. (Sorry, I don’t speak English well)
i) Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parentheses.
ii) Then For each of the propositions write a useful denial symbolically.
iii) and give a translation into ordinary English.
.
a) Not all precious stones are beautiful. (All stones)
i) $(\exists x)[(x\ is \ precious \ stone) \land \sim (x \ is \ beautiful)]$
ii) $(\forall x)[(x\ is \ precious \ stone)\Rightarrow (x \ is \ beautiful)]$
iii) All precious stones are beautiful. (All stones)
b) All precious stones are not beautiful. (All stones)
i) $(\forall x)[(x\ is\ precious \ stone) \Rightarrow \sim ( x \ is\ beautiful)]$
ii) $(\exists x) [(x\ is\ precious \ stone) \land ( x \ is\ beautiful)]$
iii) Not all precious stones are not beautiful. (All stones)
c) Some isosceles triangle is a right triangle. (All triangles)}
i) $(\exists x)[( x\ is\ isosceles\ tringle) \land (x\ is\ a\ right\ tringle)$
ii) $(\forall x) [( x\ is\ isosceles\ tringle) \Rightarrow \sim (x\ is\ a\ right\ tringle)]$
iii) For all isosceles triangle is not a right triangle. (All triangles)
d) No right triangle is isosceles. (All triangles)
i) $(\forall x)[(x\ is\ a\ right\ tringle)\Rightarrow \sim (x\ is\ isosceles)]$
ii) $(\exists x)(x\ is\ a\ right\ tringle)\land(x\ is\ isosceles)$.
iii) There exist a right triangle is not is isosceles. (All triangles)
e) All people are honest or no one is honest. (All people)
i) $(\forall x)[(x\ is\ a\ person) \Rightarrow (x\ is\ honest)] \lor (\forall x)[(x\ is\ a\ person) \Rightarrow \sim (x\ is\ honest)]$
ii) $(\exists x)[(x\ is\ a\ person)\land \sim (x\ is\ a\ honest)]\land (\exists x)[(x\ is\ a\ person)\land (x\ is\ a\ honest)]$
iii) There exist a person is not honest and there exist a person is honest. (All people)
f) Some people are honest and some people are not honest. (All people)
i) $(\exists x)[(x\ is\ preson) \land (x\ is\ honest)]\land (\exists x) [(x\ is\ preson) \land \sim (x\ is\ honest)]$
ii) $(\forall x)[(x\ is\ preson) \Rightarrow \sim (x\ is\ honest)]\lor (\forall x) [(x\ is\ preson) \Rightarrow (x\ is\ honest)]$
iii) All people are not honest or all people are honest. (All people)
g) Every nonzero real number is positive or negative. (Real numbers)
i) $(\forall x)[((x\in \mathbb{R})\land (x\neq 0))\Rightarrow (x>0)\lor (x<0)]$
ii) $(\exists x) [((x\in \mathbb{R})\land (x\neq 0)) \land ((x\leq 0)\land (x \geq 0)) ]$
or, $(\exists x) [((x\in \mathbb{R})\land (x\neq 0)) \land ((x< 0)\land (x > 0)) ]$
what is the true?
iii) There exist nonzero real number is negative and positive.
or, There exist nonzero real number is non-positive and non-negative.
What is the true?
h) Every integer is greater than −4 or less than 6. (Real numbers)
i) $(\forall x) [(x\in \mathbb{Z})\Rightarrow(x>-4)\lor (x<6)]$
ii) $(\exists x)\bigg[(x \in \mathbb{Z})\land [(x\leq -4)\land (x\geq 6)]\bigg ]$
iii) There exist integer number is less than or equal to -4 and greater than or equal to 6.
i) Every integer is greater than some integer. (Integers)
i) $(\forall x) [(x \in \mathbb{Z})\Rightarrow [(\exists y)(y \in \mathbb{Z})\land (x>y))]]$
ii) $(\exists x)\bigg[(x \in \mathbb{Z})\land [(\forall y) (y \in \mathbb{Z})\Rightarrow (x\leq y)]\bigg]$.
iii) There exist an integer number is small or equal to every integer. (Integers)
j) No integer is greater than every other integer. (Integers)}
i) $\sim (\exists x)(x\in \mathbb{Z})\land (\forall y)(y\in \mathbb{Z})\Rightarrow (x>y)$
or $(\forall x)(x\in \mathbb{Z})\Rightarrow (\exists y)(y\in \mathbb{Z})\land (x\leq y)$
ii) $(\exists x)\bigg[(x \in \mathbb{Z})\land [(\forall y) (y \in \mathbb{Z})\Rightarrow (x>y)]\bigg]$
iii) here exist an integer is greater than every other integer.
k) Between any integer and any larger integer, there is a real number. (Real numbers)
i) $[(\forall x)(x\in \mathbb{Z})\land (\forall y)(y\in \mathbb{Z}\land x<y)]\Rightarrow (\exists z)(z\in \mathbb{R})\land (x<z<y) $
ii) $(\forall x)[(x\in \mathbb{Z})\land (\forall y) ((y\in \mathbb{Z})\land x<y)]\land (\forall z)[(z \in \mathbb{R})\Rightarrow (x\geq z \geq y)]$
iii) how can I write it?
l) There is a smallest positive integer. (Real numbers)}
i)$(\exists x)[(x\in \mathbb{Z})\land (x>0) \land ((\forall y)(y \in \mathbb{Z})\land (y>0) \Rightarrow (x \leq y))]$
ii) $(\forall x)(x\in \mathbb{Z}\Rightarrow (x\geq 0))\lor ((\exists x)(y\in \mathbb{R})\land(x>y))$
iii) There is a greatest non-positive integer.
m) No one loves everybody. (All people)
i) $(\forall x) [(x\ is\ a\ preson)\Rightarrow [(\exists y)(y\ is\ a\ preson)\land (\sim(x\ loves\ y))]] $
ii) $(\exists x)\bigg[(x\ is\ a\ person)\land [(\forall y)(y\ is\ a\ person)\Rightarrow (x\ loves\ y)]\bigg]$
iii) Some people loves every body. (All people)
n) Everybody loves someone. (All people)}
i)$(\forall x)[(x \ is \ a \ person)\Rightarrow (\exists y)(y\ is\ a\ person)\land ( x\ loves\ y)]$
ii) $(\exists x)\bigg[(x\ is\ a\ person)\land [(\forall y)(y\ is\ a\ person)\Rightarrow \sim (x\ loves\ y)]\bigg]$
iii) Some people don't loves any body.
o) For every positive real number $x$, there is a unique real number $y$ such that $2^y=x$ (Real numbers)
i) $(\forall x)[(x\in \mathbb{R})\land (x>0)\Rightarrow [(\exists ! y)(y\in \mathbb{R})\land (2^y=x)]]$
ii) $(\exists x)\big[((x \in \mathbb{R})\land (x>0))\land [\sim (\exists ! y)(y\in \mathbb{R})\land (2^y=x)] \big]$
or, $(\exists x)\bigg[((x \in \mathbb{R})\land (x>0))\land [[(\exists y)(y\in \mathbb{R})(y\ is\ not\ unique)\land (2^y=x)]\lor [\sim (\exists y)(y\in \mathbb{R})\land (2^y=x)]\lor [(\forall y)(y\in \mathbb{R})\Rightarrow (2^y=x)]]\bigg]$
or, $(\exists x)\bigg[((x \in \mathbb{R})\land (x>0))\land [[(\exists y)(y\in \mathbb{R})(y\ is\ not\ unique)\land (2^y=x)]\lor [(\forall y)(y\in \mathbb{R})\Rightarrow (2^y\neq x)]\lor [(\forall y)(y\in \mathbb{R})\Rightarrow (2^y=x)]]\bigg]$
iii) There exist positive real number x, and there is a real number "not unique" y such that $2^y=x$, or for every a real number y then $2^y\neq x$, or for every a real number y then $2^y= x$.
is that true please? Thanks.
Best Answer
Good work, well done!
A small issue is that, sometimes like in i), j), k), m), n) and o), you forget $\land$.
In k), I'm sure you mean $x<z<y$.
Otherwise everything else sounds good to me. Well done!
Edit: (After including questions ii) and iii))
In g)ii), both answers you gave are correct. The first is a direct application of what you know about negative statements The second is still obtained from the first based on the fact that $x\in\mathbb{R}$ and $x\neq 0$.
I also am not sure about your translation in g)iii). I guess you know that negative means $x<0$ while nonpositive means $x\le 0$.
k)ii) is wrong.Just do the same thing than earlier to k)ii). After this,
l)ii) is wrong. The negation of $x>0$ when $x\in\mathbb Z$ is $x\le 0$. Also it's $\exists y$ that should follow, not $\exists x$. Finally, don't throw away $y<0$.
For o), it will help to know the following: if $\varphi(x)$ is a formula, $$\exists! x \varphi(x)$$ is equivalent to $$\exists x (\varphi(x) \land(\forall y (\phi(y)\implies y=x)))$$.