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.)

 

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