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does $B - (A \cup C) = B \cup (A' \cup C')$?

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$\cup\setminus$ doesn't make sense. –  martini Oct 30 '12 at 13:14
    
To render the math you use dollar signs $ and symbols like setminus are used with backslash ``. I've edited your post - is this what you originally wanted to write? –  Martin Sleziak Oct 30 '12 at 13:16

2 Answers 2

up vote 1 down vote accepted

Hint: To see that this is not true, take $$M=\{a,b,c,d,e,f,g\}$$ as the mother set and $A=\{a,b\}, B=\{a,c,d\}, C=\{c,e\}$ and evaluate both sides of your so-called equation. For giving a formal set theatrical fact use @Martin's answer.

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I thought that B -(AUC) meant exist in B, but not in A or C? –  hjggh Oct 30 '12 at 16:08
    
hmmm any trick to remember that? because the use of - here is not very intuitive –  hjggh Oct 30 '12 at 16:10
    
@MathildaPitt: Your thought was right. $B-(A\cup C)$ contains those elements which are in $B$ but not in $(A\cup C)$ –  Babak S. Oct 30 '12 at 16:15
    
yeah nevermind $B \cap U -(A \cup C) = B$ –  hjggh Oct 30 '12 at 16:18
    
@MathildaPitt: $(B\cap U)-(A\cup B)$ or $B\cap (U-(A\cup B))$? –  Babak S. Oct 30 '12 at 16:25

$B-(A\cup C)=B\cap(A\cup C)'$ (this is basically the definition)

$(A\cup C)'=A'\cap C'$ (by de Morgan)

Hence $B-(A\cup C)=B\cap A'\cap C'$.

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I thought that B -(AUC) meant exist in B, but not in A or C? –  hjggh Oct 30 '12 at 16:06
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$X\setminus Y$ denotes the relative complement or difference of two sets. It consists of elements which belong to $X$ but not to $Y$. In this case $x$ belongs to $B\setminus(A\cup C)$ if it belongs to $B$ and it doesn't belong to $A\cup C$. Which is the same thing as you wrote. An it is the same thing as saying that $x$ belongs to $B$ and to the complement of $A\cup C$, i.e. $x\in B\cap (A\cup C)'$. –  Martin Sleziak Oct 30 '12 at 16:44

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