Assertion, proof, theorem and certificate
- An assertion is any claimed statement.
- A proof is the logical argument that establishes an assertion's truth.
- A theorem is an assertion that has been proven to be true.
- A certificate is a piece of evidence for verifying an assertion.
Assertion
An assertion is a declarative statement that is claimed to be true.
All prime numbers greater than 2 are odd.
Proof
A proof is a logical argument that establishes the truth of an assertion.
A proof showing that the sum of two even numbers is always even.
Certificate
A certificate is evidence or a witness that can be used to verify the truth of
an assertion. They provide a way to efficiently check the validity of a solution
without redoing the entire proof.
In the context of a graph coloring problem, a certificate might be a specific
coloring of the graph that demonstrates it can be colored with three colors.
Theorem
A theorem is a statement that has been proven to be true based on axioms and
logical deductions. It is a specific type of assertion that is distinguished by
having a proof. Theorems are significant assertions that are established within
a formal system.
The Pythagorean Theorem, which states that in a right triangle, the square of
the length of the hypotenuse is equal to the sum of the squares of the lengths
of the other two sides.