Difference between if-then and if and only if

[inthenickoftime]

TL;DR

Author seems to use both in similar/identical context.

I found a really nice set of lectures by Herb Gross on the subject of calculus, but I'm confused by his usage of if and only if and if ... then in his supplementary notes.

It seems to me that he describes identical situations on both pages, but in one case the author uses the wording "if and only if" while on the other page, an "if ... then" expression is found. Is he using them interchangeably here? Why did he change the wording the second time around? I need to make sure I understand his usage of underlined (red) expressions otherwise I won't be able to follow more elaborate arguments that will follow.

 

[Mark44]

It seems to me that he describes identical situations on both pages, but in one case the author uses the wording "if and only if" while on the other page, an "if ... then" expression is found.

Yes, the two situations seem identical to me

Is he using them interchangeably here?

I don't think so, even if the situations seem identical. Rather, I think he's just focusing on one direction, if .. then.

Probably you know, but for statements/propositions p and q, $\text p \Leftrightarrow \text q$ means the following two implications:
$\text p \Rightarrow \text q$ and $\text q \Rightarrow \text p$. On the first page, he's talking about the iff statement. On the second page he's talking about one of the latter pair of implications to make a point. I don't believe his intent is that they are interchangeable.

 

[PeroK]

I found a really nice set of lectures by Herb Gross on the subject of calculus, but I'm confused by his usage of if and only if and if ... then in his supplementary notes.

Note that: "if $A$ then $B$"means "$A \Rightarrow B$" or "$A$ only if $B$".

And "$B \Rightarrow A$"means "$A$ if $B$".