Here's a breakdown:
* Statement: A theorem must be a clear and concise statement that can be either true or false.
* Proof: The theorem's truth must be established through a logical argument, often involving a series of steps and deductions.
* Previously established facts or axioms: The proof relies on existing knowledge, which can include other theorems, definitions, and axioms (fundamental truths assumed to be true without proof).
Examples of Theorems:
* Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
* Fundamental Theorem of Algebra: Every polynomial equation with complex coefficients has at least one complex root.
* Bayes' Theorem: A mathematical formula used to calculate conditional probabilities.
Key Characteristics of Theorems:
* Universal truth: A theorem holds true in all cases within the defined scope.
* Deductive reasoning: The proof uses logical reasoning and deduction to arrive at a conclusion.
* Building blocks of knowledge: Theorems provide fundamental truths upon which other mathematical concepts and theories are built.
In summary: A theorem is a proven statement that forms a cornerstone of mathematical understanding, providing a foundation for further exploration and discovery.