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)
Examples
Here is an example from Geometry:
Example: A Theorem, a Corollary to it, and also a Lemma!
Theorem
An inscribed angle a° is half of the central angle 2a°
Called the Angle at the Center Theorem.
Proof: Join the center O to A.
Triangle ABO is isosceles, so Angle OBA = Angle BAO = b°
And, using Angles of a Triangle add to 180°, Angle AOB = (180 - 2b)°
Triangle ACO is isosceles, so Angle OCA = Angle CAO = c°
And, using Angles of a Triangle add to 180°, Angle AOC = (180 - 2c)°
And, using Angles around a point add to 360°:
Angle BOC | = 360° - (180 - 2b)° - (180 - 2c)° |
= 2b° + 2c° | |
= 2(b + c)° |
Replace b + c with a, we get:
Angle BAC = a° and Angle BOC = 2a°
And we have proved the theorem.
(That was a "major" result, so is a Theorem.)
Corollary
(This is called The Angles Subtended by the Same Arc Theorem, but it’s really just a Corollary of The Angle at the Center Theorem)
Keeping the endpoints fixed ... ... the angle a° is always the same, no matter where it is on the circumference:
So, Angles Subtended by the Same Arc are equal.
Lemma
(This is sometimes called The Angle in the Semicircle Theorem, but it’s really just a Lemma to The Angle at the Center Theorem)
In the special case where the central angle forms a diameter of the circle:
2a° = 180° , so a° = 90°
So an angle inscribed in a semicircle is always a right angle.
(That was a "small" result, so it is a Lemma.)
Another example for you, related to Pythagoras' Theorem:
Example:
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.)