Is there a way to show that the only solution of $$\sin(x)=y$$ is $x=y=0$ with $x,y\in \mathbb{Q}$. I am seaching a way to proof it with the stuff you learn in linear algebra and analysis 1+2 (the stuff you learn in the first and second term).

Suppose $u=\frac{z(1+i)-i}{z+1}$ as $z\in \mathbb{C} \setminus{-i}$ What is the set of points $M(z)$ for which $u$ is a real number? What is the set of points $M(z)$ for which $u$ is pure imaginary number? What is the set of points $M(z)$ for which $|u|=\sqrt{2}$?

If $n > m$, then $m + k = n$ where $k$ is a natural number using the closure property under addition. Write $m + k = n$ as $k = n - m$ but we already know that $k$ is a natural number...

I am reading a research paper and in one step they have done $S = B \pmod{19}$ where $S$ and $B$ are both matrices. What does it mean? How do I calculate this?

Is there a way to show that the only solution of $$\sin(x)=y$$ is $x=y=0$ with $x,y\in \mathbb{Q}$. I am seaching a way to proof it with the stuff you learn in linear algebra and analysis 1+2 (the stuff you learn in the first and second term).