Please help me with the following question If you write it b

Please help me with the following question. If you write it by hand, make sure it is easy to read. Thank you!

Explain why this statement is false: Although 2000 years of efforts to prove the parallel postulate as a theorem in neutral geometry have been unsuccessful, it is still possible that someday some genius will succeed in proving it.

Solution

Theorem: A theorem is a mathematical statement that is proved using mathematical reasoning. It’s a formula or statement that can be proved from other formulas or statements.

Eg: Pythagorean Theorem: in a right triangle, square of the hypotenuse is equal to sum of the squares of two other sides.

Postulate: A statement, also known as an axiom, which is taken to be true without proof. Postulates are the basic building blocks from which theorems are derived.

Eg: Through two point there is one line.

In olden times, an axiom was something self-evident & supported by authority, from which other ideas could be derived. They represent basic idea, a foundational building block, something which has nothing else underlying it, upon which results may be based. Nowadays basic ideas are often postulated, to see what would happen if they were true. Postulates always were ideas which may or may not exist, but which sometimes attain acceptance. They doesn\'t have a proof. They are in existence because they doesn\'t have any contradiction.

Whereas Theorems have a proof. In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms You cannot have a theorem without at least two axioms(postulates).

In simpler words, a postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven. Both are contradictory to each other . So the statement that someday someone may prove postulate as a theorem will be always false irrespective of efforts made earlier.