What's the relationship between mathematical logic and just plain old "logic"? The notation differs, do the concepts? The rules? Background
I'm trying to cross the bridge into higher level mathematics and computer science, but learning to think mathematically has been a challenge for me.  I understand it relies heavily on concepts from logic so I've decided to learn logic concurrently.  I'm using Gensler's 3rd ed. of Introduction To Logic.  I find his software and methods helpful for self-instruction.  I'm also making my way through Rosen's Discrete Mathematics and its Applications 7th ed., and have read and continue to reference Hammack's Book of Proof.  
The Source(s) of My Confusion
Various proof techniques rely on classic logical forms, like the conditional, or the biconditional, etc.  So to sharpen my ability to work with more abstract mathematical ideas and objects I thought grounding myself in formal reasoning would help me out.  But I noticed that some things are different.  Notation is the first thing.  So Gensler uses $\supset$ for an "if ... then" statement.  Here's more examples,
Gensler $\ \ P \supset Q\ \ $  vs ( $P \implies Q\ \ $ Rosen & Hammack )
Gensler $\ \ P \equiv Q\ \ $  vs  ( $\ \ P \iff Q\ \ $ Rosen & Hammack )
Gensler $\ \ (x)Ix\ \ $       vs ( $\ \forall x P(x)\ \ $ Rosen & Hammack )
The conditional, biconditional, and simpler connectives like "and", "or" and "not," I'm fine with.  I get that "and" has many different notations.  But when it gets to quantificational logic, I get totally lost.  Here's how,
Gensler:
He develops a quantificational language by introducing the universal and existential quantifiers.  But he never mentions predicates, even though that's what he seems to be doing.  So he'll say $Ir$ for "Romeo is Italian, or $(x)Ix$ for "For all x, x is Italian".  He explains that one should "use capital letters for general terms, which describe or put in a category," (p. 182) whereas small letters should be used for singular terms which pick out a specific person or thing.  Contrast this with Rosen when explaining predicates.
Rosen:
He says let $P(x) = x > 3$.  The predicate is the "is greater than 3" part that "... refers to a property the subject of the statement can have" (p. 37),
 whereas "$x$" is the subject of the statement.  So it seems like Rosen and Gensler are using the concept the same, but one is calling it a predicate, and the other is concatenating large and small letters with quantifiers, but contextually they appear to be the same.
The reason this is confusing (yet important) to me, is that I find the math books on predicate translations and setting up a complex quantificational statement, and all of the inference rules and logical equivalences confusing.  But when I use Gensler's approach, I can translate no problem, and understand how to use the rules better.  So if I can use his approach, and it's just a notational issue, then I'd like to build my intuition that way.  So,  


*

*Am I doing this wrong? Is the trouble I'm having with inferences,
equivalences, notation, relations in logic vs math, because I'm
misapplying disparate logics? 

*Or, is this more of a "Newtonian prime vs Leibniz" difference in derivative notation type of thing?

*Does one really memorize all of those logical equivalences and inferences? Or is it kind of like trigonometry where, since there's an infinite number of identities, you're better off learning the most essential ones that can derive all of the others?

*How do you recommend I proceed? Am I just wasting my time learning this stuff (although I still want to learn the Gensler type of classical logic you'd find in philosophy), but if it's inappropriate for my immediate goal of getting better at proving math and cs ideas, and understanding higher level math, I'd rather focus my energies towards more effective strategies.  So, for example, I've started reading Smullyan's A Beginner's Guide to Mathematical Logic, and have so far found the exposition very enlightening and helpful.  But I fear once I get to the parts on logic, it's going to do what every other math proof book does in such sections and glaze over them with a bunch of unsatisfying, superficial explanations that leave me thinking, "but...why?"  Is math logic a completely different hill from the one I've been trying to climb? And should I just focus on a mathematical logic specific book like Smullyan's to achieve my goal of understanding and possibly contributing someday to higher level maths?


Thank you so much for reading this far, and for whatever help, strategies, advice, answers, etc. that you provide.  I am eternally greatful.
 A: That's a lot of questions .... but here are a few answers:
1/2.  You're doing it right. The differences you point out are indeed all just notational variants.  The specific example of $x > 3$ can be seen as the infix notation for $>(x,3)$ ... and if you don't like symbols, you could do something like $G(x,3)$ with $G(x,y)$ understood as $x > y$


*There are a lot of equivalences and inferences. And yes, you could indeed derive most of them from just some ... but in my experience it's really helpful to know more than just the elementary ones.

*Personally I think one logic book is just as fine as any other .. but if you find some material easier going, go for it! But I also agree very much with Derek's advice from the comments: read several different books, as they may all give slightly different perspectives. And even if the notational differences are just notational differences, you sometimes nevertheless look at a problem differently when it's expressed in a different notation. So, mastering a variety of notations, and being able to quickly go back and forth between them could really help you deepen your grasp of logic.
A: See if you can find a copy of J.Donald Monk ,Introduction to Set Theory,New York,McGraw Hill (1969) which I think will help answer some of your questions .The foundations of Mathematics today is in set theory using rules of inference which is called or comes from First order predicate Logic . 
  Also anything by  Smullyan is good to read .  
A: Paul Cohen was a world class mathematician who went into Logic and proved that the axiom of choice is independent of the other axioms of set theory and that the continuum hypothesis is independent of the other axioms of set theory including the axiom of choice .His method called forcing has had profound influence on mathematical logic .He was not a Logician . His work was  reported in short research notes and the book is a more readable account for mathematicians .Although particularly the early part is readable and interesting ,there is no rush to read it now and it won't help you now . also Smullyan and Fitting .Set theory and the continuum problem ,Dover ,2010 is a fantastic book on this subject but it's too advanced for you now .
  No the Set theory book by Monk (try for a used copy at Amazon or dealoz.com ) since it is probably out of print .
  However I also want to mention Murray Eisenberg ,Axiomatic theory of sets and classes ,Holt (1970) as an alternative or in addition to Monk .Eisenberg has the logic as part of the exposition through out and might be easier to get the answer to what logic mathematicians use when proving things . Good luck
