What does " => " mean in mathematics in proofs?
The task is as follows:
Verify the truth of the statement by means of direct proof:
|z| = 1 => z^n / (1 + z^2n) ∈ IR , z ∈ C
How should the arrow be interpreted, and what does it mean for the proof? It's an equation on one side and a term on the other, but what is that supposed to tell me? What should the proof reveal, and what's the best way to proceed?
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This follows:
In words, there is:
From |z ==1 follows z^n / (1 + z^2n) is an element of IR , where zε C
Or a little longer and more meaningful:
If we have a zε C, for which |z = = 1 applies, then z^n / (1 + z^2n) is an element of IR.
This is now a mathematical statement (short: sentence) and would be to prove.
This means that z=|1| lies on the unitary circle, and the other expression must be simplified to such an extent that a result without i comes out in it so that the statement is correct, right?
That’s exactly what has to come out…
Yeah, that’s right.
Thank you very much:)
That you do not know the meaning of => if you have such tasks, I see as bad joke.
Proof:
|z| = 1
=> z z_ = 1 (I write the “quer” for complex conjugation behind the z…)
=> z_ = 1 / z
=> z^n / (1 + z^2n) = z_^(-n) / (1 + z_^(-2n)
Expand with z_^(2n):
… = z_^(n) / (z_^(2n) + 1)
This is now the same as z^n / (1 + z^2n) with a large cross-section, which means that the number z^n / (1 + z^2n) is equal to its complex conjugated, thus real.
Greater or equal.
No, that would be “>=”. This sign is called “follow”.
Mathe has never been my strength. But program it.
I also do not know any language where this operator would have the same meaning in all the languages known to me.
In C# zB there is the => operator but it is used here for lambdas and expression bodies.
Although efficient software development requires similar skills as they are necessary to understand mathematics.