I'm having difficulty understanding why it is appropriate to use if and only if, something I thought I had a firm grasp on. From Lara Alcock's book, How to Study as a Mathematics Major:
Definition: A number $n$ is even if and only if there exists an integer $k$
$\quad\quad\quad\quad\,\,\,\,$such that $n=2k$.
To think about the phrase, it might be illuminating to split up this definition and write each implication separately:
$\quad$ A number $n$ is even if there exists an integer $k$ such that $n=2k$.
$\quad$ A number $n$ is even only if there exists an integer $k$ such that $n=2k$.
This is related to my point in Chapter 3 that we want our definition to "catch" the numbers that are even, and to exclude those that aren't. Can you see how?
The problem is that I don't see how and this shows that I don't truly understand iff. Can someone please answer this question?
RESPONSE TO J.-E. Pin:
Because this issue is both so important and preliminary, I've obsessed over this post and as a result cannot think about it any differently. Your answer had me the closest to understanding how to interpret iff, but I still feel as though I am missing something fundamental about two things implying one another.
Due to the fact that I've been tossing around the previous definition for a while, I think it would be beneficial to look at a different example definition from Lara Alcock's book, How to Think About Analysis.
Definition: A function $f:X\rightarrow\mathbb R$ is bounded above on X if and only if $\exists M\in\mathbb R$ s.t.$\forall x\in X,$ $\quad\quad\quad\quad\,\,\,\,f(x)\leq M.$
She goes on to say that,
[definitions] have a predictable structure, and there are two things to notice. First, each definition defines a single concept - this one defines what is means for a certain kind of function to be bounded above...Second, this term is said to apply if and only if something is true...
which in my mind means that the term applies exclusively or precisely in case something is true. How exactly does this interpretation relate to two things implying one another?