Introduction to Proof
Axioms, Conjectures and Theorems
Mathematical Statements ( https://www.toppr.com/guides/maths/introduction-to-euclids-geometry/axioms-conjectures-and-theorems/ )[br]In Mathematics, a [url=https://www.toppr.com/guides/reasoning-ability/statements/statements-and-assumptions/]statement[/url] is a sentence that has a truth value (true or false).[br]Axiom[br]Derived from the Greek word ‘Axioma’ meaning ‘true without needing a proof’, a mathematical statement assumed to be true without a proof. [br][img]https://d1whtlypfis84e.cloudfront.net/guides/wp-content/uploads/2018/02/19094526/800px-Parallel_Postulate-300x193.png[/img][br][br][br]Theorem[br]A mathematical statement that after a series of Conjectures(proof) we then know it is true.[br]So if a statement is always true without needed proof, it's called an axiom. If it needs a proof, it is called a conjecture. For a statement proven by a series of axioms, it is called a theorem.[br]
Which one is not an Axiom?
[br][i]1 Reflexive Axiom:[/i] A number is equal to itelf. (e.g a = a). [br][i]2 Symmetric Axiom:[/i] Numbers are symmetric around the equals sign. If a = b then b = a. [br][i]3 Transitive Axiom:[/i] If a = b and b = c then a = c. [br][i]4 Additive Axiom:[/i] If a = b and c = d then a + c = b + d. [br][i]5 Multiplicative Axiom:[/i] If a=b and c = d then ac = bd. [br]6 Product Axiom; if a - b = c then ac = bc
Introduction to Proof