Relative Complement is Decreasing

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Theorem

Let $X, Y, S$ be set such that $X \subseteq Y \subseteq S$


Then $\relcomp S X \supseteq \relcomp S Y$


Proof

Let $x \in \relcomp S Y$

By definition of relative complement:

$x \in S \setminus Y$

By definition of difference:

$x \in S$ and $x \notin Y$

By definition of subset:

$x \notin X$

By definition of difference:

$x \in S \setminus X$

Thus by definition of relative complement:

$x \in \relcomp S X$

$\blacksquare$


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