In this publication, we will consider what axioms, theorems and consequences are. The definitions are accompanied by relevant examples for better understanding.
What is an axiom
In order to solve many mathematical problems, very often it is required to perform certain logical actions, thanks to which one or another solution / proof can be obtained.
But there are statements in mathematics that do not require any proof.
For example:
- Through a point not on a line, there is only one line parallel to the given line.
- It is possible to draw a line through any two points, but only one.
- If the ends of two segments are combined during superposition, then the segments themselves are combined.
- Any figure is equal to itself.
These and other similar statements that do not need proof and are taken as starting points in any theory are called axioms (from ancient Greek “axiom”Meaning “situation”, “statement”). Sometimes they are also called postulates.
Axioms can be used to solve specific problems or be used to prove theorems.
Note: distortion of the formulations of the axioms and most of the theorems is not allowed, i.e. they need to be learned by heart.
What is a theorem
Unlike the axiom theorem is the proposition that needs to be proven. Those. there is some evidence for it in the theory under consideration.
For example:
- Triangle sum of angles theorem (equal to 180 degrees)
- Triangle exterior angle theorem
- Three perpendiculars theorem
There is a separate kind of so-called auxiliary theorems, which are not useful in themselves and are used only to prove other theorems. They are called lemmas (from ancient Greek “lemma”Meaning “assumption”).
For example:
If the product of several factors is divisible by a prime number p, then at least one of the factors is divisible by p (lemma Euclid).
What is a consequence
Consequence is a statement that has been derived from an axiom or theorem. And it also needs proof.
For example:
- If a line intersects one of two parallel lines, then it intersects the other.
- If two lines are parallel to a third line, then they are parallel.