Vtyjkyuk

Let $X$ and $Y$ be subsets of the universe $Omega$. Prove the following:

1) $Ysubseteq X$ iff $Xcup Y^c=Omega$

2) $Xcap Y=emptyset$ iff $X^ccup Y^c=Omega$

Here $^c$ denotes the complement

The statements do logically makes sense, however I'm having trouble proving it formally.

Let's prove the first complication.

If $Ysubseteq X$ then we have to prove that $Xcup Y^c=Omega$.

To prove it let's show both inclusions: it is clear that $Xcup Y^csubseteq Omega,$ so we only have to prove the other one. Take $xin Omega:$ if it belongs to $X$ we're done, otherwise it belongs to $X^c,$ that by hypothesis on $Y$ is a subset of $Y^c.$ So it certainly belongs to the union of $X$ and $Y^c.$

The other arrow is similar: if $Xcup Y^c=Omega$ then we have to prove that $Ysubseteq X.$

By absurd, suppose that $exists xin Ysetminus X.$ Then it wouldn't belong to $Xcup Y^c$ because it is nor in $X$ neither in $Y^c.$ So it is an element of $Omega$ which doesn't belong to $Xcup Y^c,$ which goes against the hypotheses.

The second statement is similar to prove, but if you want I'll make it.