Let P$($x$),$ Q$($x$),$ and R$($x$)$ be the statements $"$x is a professor," $"$x is Ignorant," and $"$x is vain," respectively. Express each of the following statements using quantifiers; logical connectives; and P$($x$),$ Q$($x$),$ and R$($x$),$ where the domain consists of all people

$($a$)"$No professors are ignorant."

$($b$)"$All ignorant people are vain."

$($c$)"$No professors are vain."

Determine whether $($c$)$ can be validly inferred from $($a$)$ and $($b$).$

Prove that if n is a positive integer, then n is odd if and

only if $5$n$+6$ is odd.