The definition of inductive set my textbook gave is:
A set $T$ that is a subset of the integers is an inductive set provided that for each integer $k$, if $k$ is an element in the set $T$, then $k+1$ is an element in $T$.
The book then asks if the set of odd natural numbers is an example of an inductive set.
My answer is that it is not because even though odd natural number is a subset of the integers if you add one to any of those you get an even number which is no longer within the defined set of odd natural integers. An example is $3+1$ which is $4$.
But I think this is wrong based on other definitions of inductive set that I am seeing online.
Edit: The text book is called Mathematical Reasoning Writing and Proof Version free. It is has a free download on the author's site.