Question

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) Something is not in the correct place.

b) All tools are in the correct place and are in excellent condition.

c) Everything is in the correct place and in excellent condictions.

d) Nothing is in the correct place and is in excellent condiction.

e) One of your tools is not in the correct place, but it is in excellent condition

Short answer

a) "Something is not in the correct place" is a statement into logical expressions using predicates, quantifiers, and logical connectives is express in the way of $\exists \mathbf{x}\mathbf{(}\mathbf{\neg }\mathbf{P}\left(x\right)\mathbf{)}$

b)" All tools are in the correct place and are in excellent condition”is a statement into logical expressions using predicates, quantifiers, and logical connectives of $\forall \mathit{x}\mathbf{(}\mathbf{P}\left(x\right)\mathbf{\wedge }\mathbf{R}\left(x\right)\mathbf{)}$

c)" Everything is in the correct place and in excellent condition”is a statement into logical expressions using predicates, quantifiers, and logical connectives is express in the way of $\forall \mathit{x}\mathbf{(}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}$

d)" Nothing is in the correct place and is in excellent condition”is a statement into logical expressions using predicates, quantifiers, and logical connectives is express in the way of $\mathbf{\neg }\exists \mathit{x}\mathbf{(}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}$

e)"One of your tools is not in the correct place, but it is in excellent condition” is a statement into logical expressions using predicates, quantifiers, and logical connectives is express in the way of $\mathbf{\neg }\exists \mathit{x}\mathbf{(}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}$

Step-by-step solution

∃x(¬P(x))

LetP(x)mean “xis in the correct place".

∀x(P(x)∧R(x))

LetP(x)mean “xis in the correct place". LetR(x)

Mean "xis in excellent condition".

The domain of x$ is tools.

∀x(P(X)∧R(x))

LetP(x)mean “xis in the correct place". LetR(x)

Mean “xis in excellent condition"

¬∃x(P(x)∧R(x))

Let P(x) mean “x is in the correct place". Let R (x)

Mean “x is in excellent condition".

"Nothing ..." means that there does not exist a tool such that...

∃x(¬P(x)∧R(x))

LetP(x) mean "x is in the correct place". Let R(x)

Mean "x is in excellent condition"

"But" can be interpreted as "and" in this case.

The domain of x is tools.