Consider two ONTO functions $f$ and $g$ where $f:\mathbb{N}\to A$ and $g:\mathbb{N}\to B$ such that $f(n)=\lfloor n\sec^2\theta\rfloor$ and $g(n)=\lfloor n\csc^2\theta\rfloor$ for some $\theta$. If $\sec^2\theta$ is irrational, then which of the following statement(s) hold true
(A) $A\cap B=\phi$
(B) $A\cup B=\mathbb{N}$
(C) $f$ and $g$ are bijective
(D) $A\cap B$ is non-empty finite set
My Attempt
$f(n)=\lfloor n+n\tan^2\theta\rfloor=n+\lfloor n\tan^2\theta\rfloor$
Similarly $g(n)=n+\lfloor n\cot^2\theta\rfloor$
Now we have $n\tan^2\theta<n\cot^2\theta$ or $n\tan^2\theta>n\cot^2\theta$.
So as value of $n$ increases the value of $f(n)$ and $g(n)$ also increases but none of them take same value.
This I could observe by putting $n=1,2,3,4,…$.
The correct options seem to be (A),(B) and (C) but I am not able to come up with any sort of proof.
I also tried to solve the equation $f(n_1)=g(n_2)$ so that I could get values of n for which $f(n)$ and $g(n)$ can be same.
Best Answer
$r = \sec^2\theta$ and $s = \csc^2 \theta$ are irrational numbers, greater than one, with $1/r + 1/s = 1$, so that $(\lfloor n r \rfloor)_n$ and $ (\lfloor n s \rfloor)_n$ are a pair of “complementary Beatty sequences.”
Rayleigh's theorem states that each positive integer belongs to exactly one of the two sequences, i.e. (A) and (B) are true, and (D) is false. A proof of that theorem can be found in the Wikipedia article.
(C) is true because both sequences are strictly increasing.