Reduction to absurdity is often used in mathematics. To prove a proposition, we only need to prove that its negative form is not valid.
How to get the negative form of a proposition? If you have studied mathematical logic, it is easy to understand. Now you can only understand it this way:
Original proposition: The square of all natural numbers is positive.
The standard form of the original proposition: arbitrary x, (if x is a natural number, then x? Is a positive number)
"Any" is a determiner, "X is a natural number" is a condition, "X? Is a positive number "is the conclusion. To deny a proposition, we need to deny its qualifier and conclusion at the same time. The qualifiers "arbitrary" and "existence" are mutually negative.
Negative form: not (arbitrary x, (if x is a natural number, then x? Is a positive number) = x exists, (if x is a natural number, then x? Not a positive number)
In other words, the square of at least one natural number is not positive.
And the negative proposition of a proposition is used less. Whether a proposition holds or not has nothing to do with whether its negative proposition holds or not.
It is easy to get a negative proposition of a question. Just deny all qualifiers, conditions and conclusions.
Original proposition: The square of all natural numbers is positive.
The standard form of the original proposition: arbitrary x, (if x is a natural number, then x? Is a positive number)
No proposition: there is x, (if x is not a natural number, then x? Not a positive number)
In other words, there is an unnatural number whose square is not positive.
In addition, for the inverse proposition, first deny the qualifier, and then exchange conditions and conclusions.
The inverse proposition of the proposition in the topic is: there is X, (if X? Is a positive number, then x is a natural number)
A negative proposition is a negative proposition of a negative proposition, or a negative proposition of a negative proposition, that is, the qualifier is unchanged and the conditions and conclusions are negatively exchanged.
The negative proposition of the proposition in the topic is: arbitrary x, (if x? If it is not a positive number, then x is not a natural number)