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 = m^{2} – n^{2}
- b = 2mn
- c = m^{2} + n^{2}
then a^{2} + b^{2} = c^{2}
Proof:
a^{2} + b^{2} | = (m^{2} – n^{2})^{2} + (2mn)^{2} | |
= m^{4} – 2m^{2}n^{2} + n^{4} + 4m^{2}n^{2} | ||
= m^{4} + 2m^{2}n^{2} + n^{4} | ||
= (m^{2} + n^{2})^{2} | ||
= c^{2} |
(That was a "major" result.)
Corollary
a, b and c, as defined above, are a Pythagorean Triple
Proof:
From the Theorem a^{2} + b^{2} = c^{2}, 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 = 2^{2} – 1^{2} = 4 – 1 = 3
- b = 2 × 2 × 1 = 4
- c = 2^{2} + 1^{2} = 4 + 1 = 5
(That was a "small" result.)