- #1
kntsy
- 82
- 0
Can i use the strong form every time? Why do people still use the ordinary form?
The ordinary form of the principle of mathematical induction states that if a statement is true for the first natural number, and if it is true for any given natural number, then it must also be true for the next natural number. The strong form of the principle adds the additional requirement that not only must the statement be true for the next natural number, but it must also be true for all natural numbers that come after it.
The strong form of the principle of mathematical induction is important because it allows us to prove more complex mathematical statements. It provides a stronger guarantee that a statement is true for all natural numbers, rather than just a guarantee that it is true for the next natural number.
No, the strong form of the principle of mathematical induction can only be used to prove statements that have the form "for all natural numbers". It cannot be used to prove statements about real numbers or other types of numbers.
The strong form of the principle of mathematical induction is used as a proof technique. It is often used to prove statements about natural numbers, such as divisibility or properties of sequences. The proof typically involves showing that the statement is true for the first natural number, and then showing that if it is true for any given natural number, it must also be true for the next natural number and all natural numbers after that.
Yes, there are extensions of the strong form of the principle of mathematical induction that can be used for other types of numbers, such as real numbers or complex numbers. However, these extensions are more complex and require additional mathematical tools, such as the Axiom of Choice, to prove.