Theorems, Corollaries, Lemmas
What are all those things? They sound so impressive!
Well, they are basically just facts: some result that has been arrived at.
- A Theorem is a major result
- A Corollary is a theorem that follows on from another theorem
- A Lemma is a small results (less important than a theorem)
Like this:
Example: Here is a Theorem, a Corollary to it, and also a Lemma!
Theorem:
If m and n are any two whole numbers and
- a = m2 – n2
- b = 2mn
- c = m2 + n2
then a2 + b2 = c2
Proof:
| a2 + b2 | = (m2 – n2)2 + (2mn)2 | |
| = m4 – 2m2n2 + n4 + 4m2n2 | ||
| = m4 + 2m2n2 + n4 | ||
| = (m2 + n2)2 | ||
| = c2 |
(That was a "major" result.)
Corollary
a, b and c, as defined above, are a Pythagorean Triple
Proof:
From the Theorem a2 + b2 = c2, so a, b and c are a Pythagorean Triple
(That result "followed on" from the previous Theorem.)
Lemma
If m = 2 and n = 1, then we get the Pythagorean triple 3, 4 and 5
Proof:
If m = 2 and n = 1, then
- a = 22 – 12 = 4 – 1 = 3
- b = 2 × 2 × 1 = 4
- c = 22 + 12 = 4 + 1 = 5
(That was a "small" result.)