What's the name of this algebraic property? [Complementary Subgroup Test] I'm looking for a name of a property of which I have a few examples:
$(1) \quad\color{green}{\text{even number}}+\color{red}{\text{odd number}}=\color{red}{\text{odd number}}$
$(2) \quad \color{green}{\text{rational number}}+\color{red}{\text{irrational number}}=\color{red}{\text{irrational number}}$
$(3) \quad\color{green}{\text{algebraic number}}+\color{red}{\text{transcendental number}}=\color{red}{\text{transcendental number}}$
$(4) \quad\color{green}{\text{real number}}+\color{red}{\text{non-real number}}=\color{red}{\text{non-real number}}$
If I were to generalise, this, I'd say that if we partition a set $X$ into two subsets $S$ and $S^c=X\setminus S$, then the sum of a member of $S$ and a member of $S^c$ is always in either $S^c$ or $S$.
My question is:
"Is there a name for this property (in these four cases) and is this property true in general?"
Also, does anyone have any more examples of this property?

Don't prime numbers offer a counter-example to the general truth of this property?
Prime $+$ not-prime $=$ not-prime $==> 17 + 4 = 21$
Prime $+$ not-prime $=$ prime $==> 7+4=11$
