What is the difference between a Definition and a Theorem?

Question

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.

Answer 1

A theorem provides a sufficient condition for some fact to hold, while a definition describes the object in a necessary and sufficient way.

As a more clear example, we define a right angle as having the measure of $\pi/2$. So now we can safely interchange the notion of right angle and having an angle of this exact measure.

On the other hand, we state a theorem, that $f\in C^1[0,1]$ is a Lipschitz function. It's clear that not all Lipschitz functions are $C^1$.

a)says that four right angles is equivalent that the object is a rectangle, i.e. $A(x)\iff R(x) $

b) says only that having four right angles is a sufficient condition to be a rectanlge, i.e. $A(x)\Rightarrow R(x)$

Answer 2

A definition describes the exact meaning of something, whereas a theorem proves something.

Answer 3

A definition introduces new expressions to your language. That is, if the terms "quadrilateral" and "right angle" are known, you can introduce the new notion of "rectangle" with a definition as above.

A theorem on the other hand involves known (defined) notions and expresses a provable statement about these. For example: A quadrilateral is a rectangle if and only if it is a parallelogram with equal diagonals.

By the way, often enough a theorem gives rise to a definition, a la

Theorem 1. For a plane quadrilateral the following are equivalent: (i) All four angles are right angles (ii) It is a parallelogram and the diagonals have equal length.

Proof: Yada yada yada. $_\square$

This inspires us to the

Definition. A quadrilateral satisfying one (and hence both) of the conditions in theorem 1 is called a "rectangle".

Answer 4

A definition is a syntactic thing. You can think of it as a macro that replaced a string for another string. It's a way to make things easier by shortening the amount of text we need to write. For instance, the definition of limit, as in $\lim_{x\to x_0}f(x)=L$ is short hand for "for all $\epsilon >0$ there exists $\delta >0$ such that if $|x-x_0|<\delta $ then $|f(x)-L|<\epsilon$". Another example would be the definition that $n$ is even if there exists $k\in \mathbb Z$ such that $n = 2\cdot k$. Formally speaking, definitions are not necessary. We can agree to never use definition and resort to spelling out each and every thing we talk about (excluding the meta-language, which we can't define ad-infinitum of course).

Theorems are completely different. A theorem is a creature of semantics. It relates to truth. In technical terms, a theorem is a statement, that is it is a well-formed-formula without free variables, that was proved to be correct. Informally, a theorem is some combination of letters in the formal language that is saying something about some terms. For instance, the statement $\forall x \exists y \quad x=2\cdot y$, in the universe of the integers for instance is trying to say that all integers are even, and is thus a false statement. The statment $\forall x (\exists y \quad x = 2y )\vee (\exists y \quad x = 2y+1)$ is saying that every integer is either even or odd, a true statement and thus a theorem.

Answer 5

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