1. ## Predicate and Quantifier

I have two of these question which I need help to solve:

1. Transalate this statements into logical xpression using predicates, quantifiers, and logical connectives

One of your tools is not in the correct place, but it is in an excellent condition.

I also have other question for nested quantifier:

Let Q(x,y) be the statement, student x has been a contestant on quiz show y. Express these sentences in terms of Q(x,y), quantifiers, and logical connectives, where the domain for x consists of all students at your school and for y consists of all quiz shows on television.
a. There is a student at your school whi has been a contestant on Jeopardy and on Wheel of Fortune
b. At least two students from your school hasve been contestants on Jeopardy.

2. Originally Posted by EquinoX
IOne of your tools is not in the correct place, but it is in an excellent condition.
T(x) is “x is your tool”.
C(x) is ‘x is in correct place”.
E(x) is “x is in excellent condition”.
$\displaystyle \left( {\exists x} \right)\left[ {T(x) \wedge \sim C(x) \wedge E(x)} \right]$

3. Originally Posted by EquinoX
Let Q(x,y) be the statement, student x has been a contestant on quiz show y. Express these sentences in terms of Q(x,y), quantifiers, and logical connectives, where the domain for x consists of all students at your school and for y consists of all quiz shows on television.
a. There is a student at your school whi has been a contestant on Jeopardy and on Wheel of Fortune
b. At least two students from your school hasve been contestants on Jeopardy.
S(x) is “x is a student in your school; E(x,y) ‘x equals y’; J “is Jeopardy”; W “is wheel of Fortune.”
a) $\displaystyle \left( {\exists x} \right)\left[ {S(x) \wedge Q(x,J) \wedge Q(x,W)} \right]$

b) $\displaystyle \left( {\exists x} \right)\left( {\exists y} \right)\left[ {S(x) \wedge S(y) \wedge \sim E(x,y) \wedge Q(x,J) \wedge Q(y,J)} \right]$

4. Originally Posted by Plato
T(x) is “x is your tool”.
C(x) is ‘x is in correct place”.
E(x) is “x is in excellent condition”.
$\displaystyle \left( {\exists x} \right)\left[ {T(x) \wedge \sim C(x) \wedge E(x)} \right]$
I am not sure with this answer, because the question mentions "one of your", this means that it's exactly one right? if I use there exist at least one then it means that there may be more than one

5. Originally Posted by EquinoX
"one of your", this means that it's exactly one right? if I use there exist at least one then it means that there may be more than one
Why would you say that? One and only one means exactly one.

If we say there exist an object that object is one of your tools not in the correct place but in excellence condition, then we are saying at least one object of yours fits that description

6. Originally Posted by Plato
S(x) is “x is a student in your school; E(x,y) ‘x equals y’; J “is Jeopardy”; W “is wheel of Fortune.”
a) $\displaystyle \left( {\exists x} \right)\left[ {S(x) \wedge Q(x,J) \wedge Q(x,W)} \right]$

b) $\displaystyle \left( {\exists x} \right)\left( {\exists y} \right)\left[ {S(x) \wedge S(y) \wedge \sim E(x,y) \wedge Q(x,J) \wedge Q(y,J)} \right]$
Do I have to assume that:

S(x) is “x is a student in your school ??

Can I just use x and modify the predicates??