does $B - (A \cup C) = B \cup (A' \cup C')$?
Are these two set theory statements equivalent?
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elementary-set-theory
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1$\cup\setminus$ doesn't make sense. – 2012-10-30
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0To 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? – 2012-10-30
2 Answers
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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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0I thought that B -(AUC) meant exist in B, but not in A or C? – 2012-10-30
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0hmmm any trick to remember that? because the use of - here is not very intuitive – 2012-10-30
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0@MathildaPitt: Your thought was right. $B-(A\cup C)$ contains those elements which are in $B$ but not in $(A\cup C)$ – 2012-10-30
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0yeah nevermind $B \cap U -(A \cup C) = B$ – 2012-10-30
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0@MathildaPitt: $(B\cap U)-(A\cup B)$ or $B\cap (U-(A\cup B))$? – 2012-10-30
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0the former B $\cup$ U gives B right? – 2012-10-30
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0@MathildaPitt: The union of the mother set $U$ and any subset of it, say $B$ is always the whole set $U$, but the intersection is always the small set. – 2012-10-30
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$B-(A\cup C)=B\cap(A\cup C)'$ (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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0I thought that B -(AUC) meant exist in B, but not in A or C? – 2012-10-30
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1$X\setminus Y$ denotes the [relative complement](http://en.wikipedia.org/wiki/Complement_%28set_theory%29) 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. $x\in B\cap (A\cup C)'$. – 2012-10-30