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 condition.

Answer 1

Answer:

Let P(x) = x is in the correct place

Let Q(x) =  x is in the excellent place

R(x) denotes the tool

Explanation:

a) Something is not in the correct place.

P(x) is that x is in the correct place so negation of ¬P(x) will represent x is not in the correct place. ∃x is an existential quantifier used to represent "for some" and depicts something in the given statement. This statement can be translated into logical expression as follows:

                                                    ∃x¬P(x)

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

R(x) represents the tool, P(x) represents x is in correct place and Q(x) shows x is in excellent place. ∀ is used to show that "all" tools and ∧ is used here because tools are in correct place AND are in excellent condition so it depicts both P(x) and Q(x). This statement can be translated into logical expression as follows:

                                       ∀ x ( R(x) → (P(x) ∧ Q(x))

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

Here P(x) represents correct place and Q(x) represents excellent condition ∀ represent all and here everything. ∧  means that both the P(x) and Q(x) exist. This statement can be translated into logical expression as follows:

                                              ∀ x (P(x) ∧ Q(x)