What is the negation of this statement ? "The equation has real distinct roots"

Question

What is the negation of this statement ?

"The equation has real distinct roots"

Is it "The equation has no real distinct roots "

or "The equation has no real roots "

or Is it "The equation has no real roots, but it can have real repeated roots"

Answer 1

The sentence says that the equation has real distinct roots, i.e. "real AND distinct" rutes. So that you have a statement $$P = A \cap B$$ . So the negation would be $$\sim A \cup \sim B$$ by the Morgan Rules. Where you can set for example $A$ to be real and $B$ to be distinct. Now what are the negations of real and distinct?

You could say just not real as being the negation of real, although you could take the definition of a real number and negate that.

As far as I remember a the real numbers form an ordered field, with the property of the supremum (or infimum) of any subset of it, always being a real number itself. But you can find a more rigurous definition on your textbook and try to negate that

Here is the definition and here theres a brief discussion about the negation of the real numbers https://www.quora.com/What-is-the-opposite-of-real-number

In my opinion you can only talk of $not-real$ numbers in your example. Indeed, it is not obvious (maybe you could try to prove it?) That the imaginary numbers are the negation of the real numbers. You should take the whole definition of real numbers and carefully negating each part of it. If you can then see what that set (will it even be a set?) Represent.

For instance at the beginning you should get that is should be a "not-field" or not-ordered. However again, you should go back and try to negate their definitions. My worry is that even if you actually went back to first principles and tried to negate everything, you'd end up with a contraddiction or with a set containing nothing as anything in it would contraddicts at least a couple of properties, but then again, these are just suppositions.