What givens are
$$2|A| = 3|B| = 6|A\cap B|$$
$$|A\cup B| = 28$$
I want to find
$$|A - B| = ?$$
Might I get help? I'm so confused right now and don't know where to start.
Thanks in advance!
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$\begingroup$ What is $s$ ? "size" ? $\endgroup$– Mauro ALLEGRANZANov 5, 2017 at 14:51
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$\begingroup$ What do you mean? I mean $s$ = the sum of element numbers. Should I remove that? $\endgroup$– CargobobNov 5, 2017 at 14:54
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1$\begingroup$ Is it clear now? I've edited. $\endgroup$– CargobobNov 5, 2017 at 14:55
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$\begingroup$ I have replaced your symbology with the standard symbol for the size of a set. $\endgroup$– Stella BidermanNov 5, 2017 at 14:57
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1$\begingroup$ See Inclusion–exclusion principle. $\endgroup$– Mauro ALLEGRANZANov 5, 2017 at 14:58
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3 Answers
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Let $x=|A\cap B|$. Then $$|A\cup B|=|A|+|B|-|A\cap B|.$$
$|A∪B|=28$ . Since $2|A|=3|B|=6|A∩B|=6x$, we must have that $|A|=3x$ and $|B|=2x$. This is because $2|A|=6x$, which gives $|A|=3x$ and $3|B|=6x$, which gives $|B|=2x$.
Therefore $$28=3x+2x-x\,.$$
You will find $x=7$. Thus $|A|=21$, $|B|=14$ and $|A\cap B|=7$. So $|A-B|=14$.
answered Nov 5, 2017 at 15:00
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$\begingroup$ Can you be more clear at $|A\cup B|=|A|+|B|-|A\cap B|$ this part? I couldn't get how we found $x=7$ perfectly. $\endgroup$– CargobobNov 5, 2017 at 15:10
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$\begingroup$ $|A\cup B|=28$. Since $2|A|=3|B|=6|A\cap B|=6x$, we must have that $|A|=3x$ and $|B|=2x$. So $28=3x+2x-x$. Does that make sense? $\endgroup$ Nov 5, 2017 at 15:11
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1$\begingroup$ We have $2|A|=3|B|=6|A\cap B|=6x$. Since $2|A|=6x$, $|A|=3x$, and since $3|B|=6x$, $|B|=2x$. $\endgroup$ Nov 5, 2017 at 18:56
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1$\begingroup$ You are great! I got it now. Can you update your answer? It would be good if you can add this into your answer. $\endgroup$– CargobobNov 5, 2017 at 18:56
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Use $$ |A\cup B|=|A|+|B|-|A\cap B|$$ to find $|A\cap B|$ (as well as $|A|,|B|$) and then $$|A-B|=|A|-|A\cap B| $$ for the final result.
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The principle of inclusion-exclusion tells us that $|A\cup B|=|A|+|B|-|A\cap B|$. Your chain equality plus the fact that $|A\cup B|=28$ allows you to then calculate the size of $A,B,$ and $A\cap B$.
Then use the fact that $|A-B|=|A|-|A\cap B|$, which follows straight from the definition of set minus.
answered Nov 5, 2017 at 15:01