SurferStrobe said:Homework Statement
Prove a theorem using direct proof, mathematical induction, contraposition, or contradiction.
Homework Equations
"If m divides n, then m <= n."
The Attempt at a Solution
(a) Suppose m divides n, then m = nk for some integer k.
(b) If k = 1, the m = n(1) = n.
That show equality part. How do I now show inequality? I'm at a loss for the next step.
The purpose of proving basic theorems is to establish the validity of fundamental mathematical principles. This helps to build a strong foundation for more complex mathematical concepts and allows for the development of new theories and applications.
The process of proving a basic theorem involves using logical reasoning and mathematical techniques to show that the theorem is true for all possible cases. This typically includes breaking down the theorem into smaller, more manageable parts and using previously established theorems and axioms as building blocks.
Proving a basic theorem rigorously ensures that it is universally applicable and not just a coincidence based on a few examples. It also allows for a deeper understanding of the theorem and its implications, as well as the ability to apply it to other areas of mathematics.
No, a basic theorem cannot be proven wrong. Once a theorem has been rigorously proven, it is accepted as a fundamental truth in mathematics. However, it is possible for a previously proven theorem to be disproven or modified by new discoveries or advancements in mathematics.
Basic theorems serve as the foundation for many practical applications in fields such as physics, engineering, and computer science. They provide a framework for understanding and solving complex problems, and can also be used to develop new technologies and innovations.