Mathematical:
* The sum of the angles in a triangle is 180 degrees. (Proven using geometry and the concept of parallel lines)
* The square root of 2 is irrational. (Proven using proof by contradiction)
* The equation x^2 + 1 = 0 has no real solutions. (Proven using the properties of real numbers and the quadratic formula)
Logical:
* All bachelors are unmarried. (Proven using the definition of "bachelor")
* If it is raining, then the ground is wet. (Proven using the concept of cause and effect)
* If A implies B, and B implies C, then A implies C. (Proven using the rules of logical deduction)
Scientific:
* Water boils at 100 degrees Celsius at sea level. (Proven through experimentation and scientific observation)
* Gravity pulls all objects towards each other. (Proven through observations and experiments)
* The Earth revolves around the Sun. (Proven through observations, calculations, and the development of models)
It's important to note that:
* Not all statements can be proven. Some statements are subjective, based on opinions or beliefs, or are simply unfalsifiable.
* Proofs can be complex and require specific methods. The type of proof needed depends on the nature of the statement.
* "Proven" doesn't necessarily mean "absolutely certain." In some cases, a statement might be considered proven based on the current state of knowledge, but future discoveries could lead to revisions or changes.
Let me know if you would like to see more examples or have any other questions!