define axioms postulates propositions and theorems with their 3 examples
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An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[1][2]
The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.[3] As used in modern logic, an axiom is a premise or starting point for reasoning.[4]
As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic).
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments
example :
Examples of axioms can be 2+2=4, 3 x 3=4 etc. In geometry, we have a similar statement that a line can extend to infinity. This is an Axiom because you do not need a proof to state its truth as it is evident in itself.
mathematical statement that we know is true and which has a proof is a theorem. We can further explain it as a series of Conjectures (proof) that combine together to give a true result. So if a statement is always true and doesn’t need proof, it is an axiom. If it needs a proof, it is a conjecture. A statement that has been proven by logical arguments based on axioms, is a theorem. We generate a theorem by the way of analysis and proof. Consider them your weapons, Superheroes as you start on this journey through the mazes of mathematics and arrive at solutions and save the day.
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