Given the predicates $P(x)$:$x$ is a penguin; $S(x)$:$x$ is a seal; $N(x)$:$x$ is nice;

Prove the following argument is valid, citing each property you use: "Jed is nice. All penguins are nice. All seals are not nice. Every animal is either a penguin or a seal. Therefore, Jed is a penguin."

$\backslash$table$[[$Step$,$Statement,Reason$],[1,N(Jed),$Premise$],[2,,$Premise$],[3,,]]$

Prove that for all positive integers $n$:$n$ is odd if and only if $n^2+2n+3$ is even.