Question

Translate each of these statements into logical expressions in three different ways by varying the domain and by using predicates with one and with two variables.
a) Someone in your school has visited Uzbekistan.
b) Everyone in your class has studied calculus and C++.
c) No one in your school owns both a bicycle and a motorcycle.
d) There is a person in your school who is not happy.
e) Everyone in your school was born in the twentieth century.

Short answer

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

$$\begin{gathered} \mathrm{a})\exists \mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{,}\exists \mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{)}\wedge \mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{}\mathbf{and}\exists \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Uzbekistan}\mathbf{)}\mathbf{.} \\ \mathrm{b})\mathbf{}\forall \mathbf{x}\mathbf{(}\mathbf{Y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\to }\mathbf{\left(}\mathbf{T}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{calculus}\mathbf{)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{C}\mathbf{+}\mathbf{+}\mathbf{\left)}\mathbf{\right)}\mathbf{)}\mathbf{.} \\ \mathrm{c})\forall \mathbf{x}\mathbf{\left(}\mathbf{Y}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Bicycle}\mathbf{)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Motorcycle}\mathbf{)}\mathbf{\left)}\mathbf{\right)}\mathbf{.} \\ \mathrm{d})\exists \mathbf{xY}\mathbf{(}\mathbf{x}\mathbf{)}\wedge \mathbf{\neg }\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Happy}\mathbf{\left)}\mathbf{\right)}\mathbf{.} \\ \mathrm{e})\forall \mathbf{x}\mathbf{(}\mathbf{Y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\to }\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{20}\mathbf{)}\mathbf{)}\mathbf{.} \end{gathered}$$

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 variable x has truth value.

Finding translated form into logical expression for statement a

Let’s take domain consists of students in the class. Assume Q(x) can be “x has visited Uzbekistan”. When that only schoolmates have visited Uzbekistan, then the logical expression is $\exists \mathrm{xP}\left(\mathrm{x}\right)$. When domain have everyone visited Uzbekistan, then the expression is $\exists \mathrm{xP}\left(\mathrm{x}\right)\wedge \mathrm{Q}\left(\mathrm{x}\right))$. Assume that person has gone to visit country be T(x,y). The statement can be translated as$\exists \mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{,}\exists \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{}\mathbf{and}\exists \mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Uzbekistan}\mathbf{)}\mathbf{.}$

Finding translated form into logical expression for statement b

Let’s take x be the student and the propositional function P(x) and x knows calculus and Q(x) knows C++. When that only schoolmates have studied, then the logical expression is$\forall \mathrm{x}\left(\mathrm{P}\right(\mathrm{x})\wedge \mathrm{Q}(\mathrm{x}\left)\right)$. When domain have everyone has studied, then the expression is $\forall \mathrm{x}\left(\mathrm{Y}\right(\mathrm{x})\to (\mathrm{P}\left(\mathrm{x}\right)\wedge \mathrm{Q}\left(\mathrm{x}\right)\left)\right)$. Assume that person has gone to visit country be T(x,y). The statement can be translated as$\forall \mathbf{x}\mathbf{\left(}\mathbf{Y}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{calculus}\mathbf{)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{C}\mathbf{+}\mathbf{+}\mathbf{)}\mathbf{\left)}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement c

Let’s take x be the student and the propositional function P(x) and x knows calculus and Q(x) knows C++. When that only schoolmates, then the logical expression is $\forall \mathrm{x}(\neg \mathrm{P}(\mathrm{x})\wedge \mathrm{Q}(\mathrm{x}\left)\right)$. When domain have everyone has studied, then the expression is $\forall \mathrm{x}\left(\mathrm{Y}\right(\mathrm{x})\to \neg (\mathrm{P}\left(\mathrm{x}\right)\wedge \mathrm{Q}\left(\mathrm{x}\right)\left)\right)$. Assume that person has gone to visit country be T(x,y). The statement can be translated as$\forall \mathbf{x}\mathbf{\left(}\mathbf{Y}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Bicycle}\mathbf{)}\wedge \mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Motorcycle}\mathbf{)}\mathbf{\left)}\mathbf{\right)}\mathbf{.}$

Finding translated form into logical expression for statement d

Let’s take x be the student and the propositional function P(x) and x be happy. When that only schoolmates, then the logical expression is $\exists \mathrm{x}\neg \mathrm{P}\left(\mathrm{x}\right)$. When domain have everyone has studied, then the expression is $\exists \mathrm{x}\left(\mathrm{Y}\right(\mathrm{x})\wedge \neg \mathrm{P}(\mathrm{x}\left)\right)$. Assume that person has gone to visit country be T(x,y). The statement can be translated as$\exists \mathbf{xY}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\wedge \mathbf{\neg }\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{Happy}\mathbf{)}\mathbf{)}\mathbf{.}$

Finding translated form into logical expression for statement e

Let’s take x be the student and the propositional function P(x) and x born in twentieth century. When that only schoolmates, then the logical expression is $\forall \mathrm{xP}\left(\mathrm{x}\right)$. When domain have everyone has studied, then the expression is$\forall \mathrm{x}\left(\mathrm{Y}\right(\mathrm{x})\to \mathrm{P}(\mathrm{x}\left)\right)$. Assume that person has gone to visit country be T(x,y). The statement can be translated as$\forall \mathbf{x}\mathbf{\left(}\mathbf{Y}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{20}\mathbf{\left)}\mathbf{\right)}\mathbf{.}$