Here is my Discrete Math question
Let $P(n)$ denote the statement "$n$ is prime." Find the truth value of "very $n$ in empty set is a prime number". And explain your answer.
I understand every $n$ in empty set is NOT a prime number. And what does it mean by "Find the truth value of" in this question?
Professor gave me this hint but I don't understand.
Hey this is a hint. What is the truth value of NOT every $n$ in empty set is a prime number?
On
To find the truth value of a statement is to discover whether the statement is true or false. So suppose "for all $n$ in $\varnothing$, $n$ is prime" is false. Then there must be an $n$ in $\varnothing$ which is not prime. Can you point me to such an $n$?
On
The "truth value" of a statement is just whether it is true or false. So the problem is asking you to determine whether the statement is true or false.
On
"Let P(n) denote the statement n is prime" is a statement which does not appear other times in the premise nor in the question, so we will totally ignore it.
"for all n in empty set, n is prime number" is a statement which does not depend by n, so it could be true or false. In this case, it's true because you are checking a property over the empty set (google: vacuous truth).
Your professor hint makes sense, because it makes the logic way more evident. The negation of "for all n in empty set, n is prime number" is
"NOT for all n in empty set, n is prime number"
which is equivalent to
"exists an n in empty set, n is NOT prime number".
This is evidently false, because it doesn't exist any element in the empty set, with or without the stated property.
The statement "for all $x\in S, P(x)$" is true if $S=\emptyset$, no matter what the proposition $P$ is.
So, every integer in $\emptyset$ is prime, as well as every integer in $\emptyset$ is composite, as well as every integer in $\emptyset$ is equal to itself, and to $\pi$, and every unicorn in $\emptyset$ is rainbow-coloured. Some of those statements sound nonsensical, but are nevertheless true.
This is because the opposite of "for all $x\in S, P(x)$" is "there exists $x\in S$ such that $\lnot P(x)$". This is obviously false if $S=\emptyset$, because there doesn't exist an $x\in\emptyset$ at all, let alone such $x$ for which $\lnot P(x)$ would be true.