$(1).$ 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$)$

$(2).$ Prove that if $n$ is a positive integer, then $n$ is odd if and only if $5n+6$ is

odd