How can this inverse of conditional statement be equivalent? "A positive integer is a prime only if it has no divisors other than one and itself." The inverse of this conditional statement is : " A positive integer is not prime if it has divisors other than one and itself."
In logic the inverse isn't equivalent to the original statement. Can someone please clarify it for me??
 A: I believe you are going wrong with words. From Rosen's Discrete Mathematics, $p \rightarrow q$, can be expressed this way:


*

*$p$ only if $q$.

*$q$ if $p$.
It means that if your first statement is $p \rightarrow q$, your second statement is actually $\neg q\rightarrow \neg p$ (contrapositive), not $\neg p\rightarrow \neg q$ (inverse).
A: "A positive integer is a prime if and only if it has no divisors other than one and itself." 
"A positive integer is not prime if and only if it has divisors other than one and itself."
These are equivalent in logic.
A: The statement you gave is equivalently "If a positive integer is prime, then it has no divisors other than one and itself."
The converse is "If a positive integer is composite, then it has a divisor which is neither itself nor one." 
However, in definitions, we often sloppily say "if" when we mean "if and only if". For example, you might see "a metric space is compact if every open cover has a finite subcover" as a definition of "compact", when really it is intended to be read as "a metric space is compact if and only if every open cover has a finite subcover". In the context of a definition, "if" is usually meant as "iff".
Therefore, in this instance, the converse is equivalent to the original statement, because we are working with a definition.
A: Let $n$ a positive integer, $p$ "$n$ is a prime.", and $q$ "$n$ has no divisors other than one and itself." then first sentence can be written as
$$p \rightarrow q,$$
and second sentence can be written as
$$\neg q\rightarrow \neg p.$$
They are equivalent, because $\neg q\rightarrow \neg p$ is a contrapositive of $p \rightarrow q$, not an inverse. If you want to suggest inverse, then
$$\neg p \to \neg q,$$
equivalent to "If a positive integer is not prime it has divisors other than one and itself.", or "A positive integer has divisors other than one and itself if it is not prime." 
