Prove that $A,B$ ideals in $R==(P(X),\Delta , \cap)$.

Question

If $X=${$1,2$}, and

$R=(P(X),\Delta , \cap)$

Prove that

$(A,\Delta,\cap)$

And

$(B,\Delta,\cap)$

Ideals in $R$, such that:

$A=${$\phi , ${$1$}}

$B=${$\phi , ${$2$}}

I know I should to prove

$x-y\in A$ and

$rx,xr\in A$

For every $x,y\in A$ and $r\in R$.

But i don't know, take $x=\phi$ and $y=${$1$} ? Or take it $x$ and $y$ to generally ?

Sorry, i don't speak English well.

Thanks

Answer 1

We need $x-y\in A$ for every possible selections of $x,y\in A$
But there are only 4 possibilities, and each one is trivial.

Similarly, $rx\in A$ has to hold for every possible choice of $r\in R$ and $x\in A$.