This is more interesting than it may seem.
But for now, no comment ;)
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This is more interesting than it may seem.
But for now, no comment ;)
Previous iteration: #701531
Same class of infinite number of solutions, I'd say.
That's a win.
I'll post a video tomorrow elaborating on this concept for people unfamiliar with the works of George Cantor.
Here is one of the many videos that illustrates this idea.
That's my first thought, too. Unfortunately, I won't have time to dive into today's puzzle.
yes, it has one positive and one negative. while cos(x)=0 only has one.
Are you sure there is only one solution for cos(x)=0?
both have an infinite number of solutions. cos(x^2) = 0 has broader distribution of solutions compared to the linear and equally spaced solutions of cos(x) = 0. cos(x^2) = 0 is more dense in unique values.
That's correct.
But can one argue that one infinity is larger than the other? They're both infinite...
On what domain, sir?
Infinite domain. The real number line.
I should have been clearer, indeed.
Both have an infinity of solutions. And I'm guessing they have the same cardinality of solutions?
If cos(x)=0 then let y=sqrt(x) if x>0 or y=-sqrt(-x) if x<0. Then cos(y^2)=0.
Since there's a 1:1 mapping between these x's and these y's, the two sets have the same cardinality.
I believe that both have infinite number of solutions. However, when looking into the chart of sin(x) and sin(x^2), it is quite obvious, that both the infinities have different cardinality.
I'd say that the number of solutions for cos(x)=0 has same cardinality as the set of all natural numbers. However, for the cos(x^2)=0, the cardinality would be as the set of all real numbers. At least, it is not greater.
Cos (X) = Cos (X mod 360), therefore infinite solutions.
The same applies to Cos (y), with y = x^2