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Implications of Laws for Operations on Sets

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De Morgan's Laws imply, for instance,
$\displaystyle C\setminus (A\cap C)$ $\displaystyle =$ $\displaystyle (C\setminus A)\cup (C\setminus C)\;=\;(C\setminus A)\cup \emptyset\;=\;C\setminus A\,,$  
$\displaystyle C\setminus (A\cup C)$ $\displaystyle =$ $\displaystyle (C\setminus A)\cap (C\setminus C)\;=\;(C\setminus A)\cap \emptyset\;=\;\emptyset \,.$  

It follows from the distributive laws, that
$\displaystyle (A\cup B)\cap B$ $\displaystyle =$ $\displaystyle (A\cap B)\cup (B\cap B)\;=\;(A\cap B)\cup B\,.$  

Here it is $ (A\cap B)\cup B=B$, since $ (A\cap B)\subseteq B$.
(Authors: Wipper/Abele)

see also:

  automatisch erstellt am 6.  6. 2006