Questions

Let $P(x)$ be the predicate $"x>\frac{1}{x}.$

$($a$)$ Write $P(2),P(\frac{1}{2}),P(-1),P(-\frac{1}{2}),$ and $P(-8),$ and indicate which of these statements are true and which are false.

$($b$)$ Find the truth set of $P(x)$ if the domain of $x$ is $R,$ the set of all real numbers.

$($c$)$ If the domain is the set $R^{+}$of all positive real numbers, what is the truth set of $P(x)?$

Find the truth set of each predicate.

$($a$)$ predicate: $\frac{6}{d}$ is an integer, domain: $Z$

$($b$)$ predicate: $\frac{6}{d}$ is an integer, domain: $Z^{+}$

$($c$)$ predicate: $1x^{2}4,$ domain: $R$

$($d$)$ predicate: $1x^{2}4,$ domain: $Z$

Consider the following statement: EExinR such that $x^2=2.$ Which of the following are equivalent ways of expressing this statement?

$($a$)$ The square of each real number is $2.$

$($b$)$ Some real numbers have square $2.$

$1$

$($c$)$ The number $x$ has square $2,$ for some real number $x.$

$($d$)$ If $x$ is a real number, then $x^2=2.$

$($e$)$ Some real number has square $2.$

$($f$)$ There is at least one real number whose square is $2.$

Let $D$ be the set of all students at your school, and let $M(s)$ be $"s$ is a math major," let $C(s)$ be $"s$ is a computer science student," and let $E(s)$ be $"s$ is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates $M(s),C(s),$ and $E(s).$

$($a$)$ There is an engineering student who is a math major.

$($b$)$ Every computer science student is an engineering student.

$($c$)$ No computer science students are engineering students.

$($d$)$ Some computer science students are also math majors.

$($e$)$ Some computer science students are engineering students and some are not.

$($a$)$ Write a formal negation for each of the following statements:

i$.AA$ fish $x,x$ has gills.

ii$.AA$ computers $c,c$ has a CPU.

iii. $EE$ a movie $m$ such that $m$ is over $6$ hours long.

iv$.EE$ a band $b$ such that $b$ has won at least $10$ Grammy awards.

$($b$)$ Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous.

i$.$ All dogs are friendly.

ii$.$ All people are happy.

iii. Some suspicions were substantiated.

iv$.$ Some estimates are accurate.

Rewrite the statement

$"$No good cars are cheap" in the form $"$ AAx, if $P(x)$ then $Q(x)."$ Indicate whether each of the following arguments is valid or invalid, and justify your answers.

$($a$)$ No good car is cheap.

A Rimbaud is a good car.

:$.$ A Rimbaud is not cheap.

$($b$)$ No good car is cheap.

A Simbaru is not cheap.

:$.$ A Simbaru is a good car.