Like the title has stated, there is one thing you need to learn before you go into the second year of mathematics in university. That one thing is certainly neither the first year Calculus class nor the Linear Algebra class. But the definition of the terms you will see a lot in those advanced math classes such as Analysis, Abstract Algebra, or Number Theory, etc. So here they are.

**Caution: The followings are stolen from an undisclosed source in the American Mathematical Society aka AMS. 😀**

CLEARLY:

I don’t want to write down all the “in-between” steps.

TRIVIAL:

If I have to show you how to do this, you’re in the wrong class.

OBVIOUSLY:

I hope you weren’t sleeping when we discussed this earlier, because I refuse to repeat it.

RECALL:

I shouldn’t have to tell you this, but for those of you who erase your memory tapes after every test…

WLOG (Without Loss Of Generality):

I’m not about to do all the possible cases, so I’ll do one and let you figure out the rest.

IT CAN EASILY BE SHOWN:

Even you, in your finite wisdom, should be able to prove this without me holding your hand.

CHECK or CHECK FOR YOURSELF:

This is the boring part of the proof, so you can do it on your own time.

SKETCH OF A PROOF:

I couldn’t verify all the details, so I’ll break it down into the parts I couldn’t prove.

HINT:

The hardest of several possible ways to do a proof.

BRUTE FORCE (AND IGNORANCE):

Four special cases, three counting arguments, two long inductions, “and a partridge in a pair tree.”

SOFT PROOF:

One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.

ELEGANT PROOF:

Requires no previous knowledge of the subject matter and is less than ten lines long.

SIMILARLY:

At least one line of the proof of this case is the same as before.

CANONICAL FORM:

4 out of 5 mathematicians surveyed recommended this as the final form for their students who choose to finish.

TFAE (The Following Are Equivalent):

If I say this it means that, and if I say that it means the other thing, and if I say the other thing…

BY A PREVIOUS THEOREM:

I don’t remember how it goes (come to think of it I’m not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows.

TWO LINE PROOF:

I’ll leave out everything but the conclusion, you can’t question ’em if you can’t see ’em.

BRIEFLY:

I’m running out of time, so I’ll just write and talk faster.

LET’S TALK THROUGH IT:

I don’t want to write it on the board in case I make a mistake.

PROCEED FORMALLY:

Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses).

QUANTIFY:

I can’t find anything wrong with your proof except that it won’t work if x is a moon of Jupiter (Popular in applied math courses).

PROOF OMITTED:

Trust me, It’s true.