User:Caliburn/s/mt/Horizontal Section of Set Difference is Set Difference of Horizontal Sections

From ProofWiki
< User:Caliburn‎ | s‎ | mt
Jump to navigation Jump to search

Theorem

$\paren {E_1 \setminus E_2}^y = E_1^y \setminus E_2^y$


Proof

Note that:

$x \in E_1^y \setminus E_2^y$

This is equivalent to:

$x \in E_1^y$ and $x \not \in E_2^y$.

That is:

$\tuple {x, y} \in E_1$ and $\tuple {x, y} \not \in E_2$

This is the case if and only if:

$\tuple {x, y} \in E_1 \setminus E_2$

Which is equivalent to:

$x \in \paren {E_1 \setminus E_2}^y$

So we have:

$x \in E_1^y \setminus E_2^y$ if and only if $x \in \paren {E_1 \setminus E_2}^y$.

giving:

$\paren {E_1 \setminus E_2}^y = E_1^y \setminus E_2^y$