Dulanjana
2002-12-17 15:41:24 |
Any difference
What is the difference between a theorem, axiom, corollary, and lemma? |
friedlinguini
2002-12-18 02:16:53 |
Re: Any difference
In general:
Theorem - something that is proven
Axiom - something that is simply assumed to be true (same as postulate)
Corollary - something which is trivially proven from a theorem
Lemma - something that gets proven in the course of proving a theorem |
TomM
2002-12-18 03:38:38 |
Re: Any difference
Just to expand:
An axiom (or postulate) is the basic building block on which proofs are built. It is usually, but not always intuitively obvious. Of the five axioms of Euclidean geometry, the first four are intuitively obvious; the fifth is not. Dropping or changing it leads to the creation of non-Euclidean geometries.
A theorem is a statement which has been proven. A lemma and a corollary are theorems that are special case scenarios of other theorems.
If before you can prove Theorem A, you need to prove special case B as a separate theorem, then Theorem B is a lemma of Theorem A.
If special case B is a commonly seen variant on Theorem A, (often with an additional conclusion not included in the general case), then (since most of the work was done in proving Theorem A) Theorem B is a corollary of Theorem B. |
levik
2002-12-18 05:18:08 |
Re: Any difference
Just for the uninitiated (not that I know anyone in particular :D), what ARE the five postulates of Euclidean geometry? |
friedlinguini
2002-12-18 05:42:40 |
Re: Any difference
http://mathworld.wolfram.com/EuclidsPostulates.html
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. |
Dulanjana
2002-12-18 16:55:08 |
Re: Any difference
I guess you can call those 5 postulates the rules of the game of geometry!
Now I will take the pythogorean theorem
a^2 + b^2 = c^2
a^2 = c^2 - b^2
a^2 = (c + b)(c - b)
"The product of the sum and difference of the hypotenuse and side of a right angled triangle equals the square of the third side" (Just made this up)
Will this be a corrolary? |
Dulanjana
2002-12-18 16:57:28 |
Re: Any difference
By the way is it OK to "assume" in mathmatics? Shouldnt everthing be proven right? |
friedlinguini
2002-12-19 02:59:42 |
Re: Any difference
Your statement would be a corollary of the Pythagorean Theorem.
Assumptions are generally OK as long as they're mentioned as such. Eventually, though, you have to make some assumptions (which generally wind up being your axioms). In other cases, you can assume A as long as your only goal is to prove that B is true if A is true. |
Fernando
2003-03-28 08:35:51 |
Re: Any difference
It's OK to assume things which you know HAVE BEEN PROVEN before, like in a book, in a class, or something you've proved yourself... |
