What is the correct notation for non simultaneous (asynchronous) equations?

Question

most common practice is when we type simultaneous equations in form like that: \begin{cases} x + y = 2 \\ x - y = 2 \\ \end{cases} Related question - What is the notation for a system of simultaneous equations?

But, what if have equation like $$ x^4 - 5x^2 + 6 =0 $$ and i want to denote it into 2 equations: $$x^2=2$$ and $$x^2=3$$ They both will be the solutions but not simultaneous, I mean if I say that $x=+\sqrt3$ it will be one of the solutions for 2nd and original equations, but not for 1st.

I saw notations with square brackets, like is that TeX question or with using word $ or $ between them.

So how can I correctly write such equations?

Answer 1

"$x^2=2$ or $x^2=3$" is perfectly acceptable notation.

Answer 2

Also in the first example, we can write both equations also as one equation $(x+y-2)(x-y-2)=0$. However, the system of two equations need not be equivalent to the one given equation, i.e., $x^4-5x^2+6=0$ is not equivalent to $x^2-2=0$ and $x^2-3=0$, since $(x^2-2)(x^2-3)=0$ only implies that one factor must be zero, but not both.

In general, we have here systems of (polynomial) equations. For example,

\begin{align*} xy^2z + 27xz^4 - y^5 & =0\\ x^3yz -xyz +20 & =0\\ xy^3z+xz^2+x+y+z & =0 \end{align*}

A solution has to satisfy all equations simultaneously. Your systems are \begin{align*} x+y-2 & =0,\\ x-y-2 & = 0, \end{align*} and \begin{align*} (x^2-2)(x^2-3) & =0, \end{align*}