Question

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.
a) No one is perfect.
b) Not everyone is perfect.
c) All your friends are perfect.
d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect.

Short answer

The translated form of the statement into logical expressions can be explained.

$$\begin{gathered} \mathrm{a})\forall \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{.} \\ \mathrm{b})\mathbf{\neg }\forall \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.} \\ \mathrm{c})\mathbf{}\forall \mathbf{x}\mathbf{(}\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\to \mathbf{P}\mathbf{\left(}\mathbf{X}\mathbf{\right)}\mathbf{.} \\ \mathrm{d})\mathbf{}\exists \mathbf{x}\mathbf{(}\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{.} \\ \mathrm{e})\mathbf{}\forall \mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \forall \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.} \\ \mathrm{f})\mathbf{}\mathbf{(}\mathbf{\neg }\forall \mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\vee \mathbf{(}\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{.} \end{gathered}$$

Step-by-step solution

Concept for finding domain

The value of propositional function P at x can be statement P(x). The statement will become propositional and said to be variable x has truth value.

Finding translated form into logical expression for statement a

Given statement P(x) can be propositional function i.e., x is perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\forall \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement b

Given statement P(x) can be propositional function i.e., Not everyone is perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\mathbf{\neg }\forall \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement c

Given statement P(x) can be propositional function i.e., All your friends are perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\forall \mathbf{x}\mathbf{\left(}\mathbf{Q}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{.}$

Finding translated form into logical expression for statement d

Given statement P(x) can be propositional function i.e., At least one of your friends is perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\exists \mathbf{x}\mathbf{\left(}\mathbf{Q}\mathbf{\right(}\mathbf{x}\mathbf{)}\wedge \mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement e

Given statement P(x) can be propositional function i.e., Everyone is your friend and is perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\forall \mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \forall \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement f

Given statement P(x) can be propositional function i.e., Not everybody is your friend or someone is perfect. Assume that the domain is all people. The statement can be written in form of logical expression as$\mathbf{(}\mathbf{\neg }\forall \mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\vee \mathbf{(}\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{.}$