What is the difference between a Definition and a Theorem? This may get into a discussion, but I have a homework problem and it tells me there is a difference between a definition and a theorem.  I don't know how to differentiate the two in this question:
Consider the domain of all quadrilaterals.  Let
A(x) = "x has four right angles."
R(x) = "x is a rectangle."
Write the meaning of each mathematical statement in the predicate logic, keeping in mind the logical distinction between definitions and theorems.
(a) Definition. A quadrilateral is a rectangle if it has four right angles.
(b) Theorem. A quadrilateral is a rectangle if it has four right angles.

This is more an explanation belonging to logic.


*

*a definition is a rewriting and has to be an equivalence


in your example
We don't want to write "is a rectangle" or " has four right angles."  in a formula it would make the formula to long  so we shorten it to "A" and "R" to make it abstract (and also to get rid of unwritten expectations, how do you proof that $ \forall x ( A(x) \leftrightarrow  R(x) )$ ?, it is not valid in non euclidian geometry )
Another example:
Russels principia mathematica only has the $ \lnot $  and $ \lor $ as primitive  connectives
and
$ (P \land Q ) $ is defined as $ \lnot( \lnot P \lor \lnot Q) $
$ (P \rightarrow Q ) $ is defined as $ ( \lnot P \lor  Q) $
$ (P \leftrightarrow Q ) $ is defined as $(P \rightarrow Q ) \land (Q \rightarrow P )$   
Which means that $ (P \land Q ) $ , $ (P \rightarrow Q )  $ and $ (P \leftrightarrow Q ) $ are not really there but are just shorthand for their primitive formulas.
(and also that if you see $ \lnot( \lnot P \lor \lnot Q) $   you can replace that with $ (P \land Q ) $ )
You could think of a definition as some kind of axiom (because it is also an assumed truth)
A theorem is on the other hand something that needs to be proved and mostly the result is just a one way implication.
Good luck 
