# Calculating $|A-B|$

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.

• What is $s$ ? "size" ? Nov 5, 2017 at 14:51
• What do you mean? I mean $s$ = the sum of element numbers. Should I remove that? Nov 5, 2017 at 14:54
• Is it clear now? I've edited. Nov 5, 2017 at 14:55
• I have replaced your symbology with the standard symbol for the size of a set. Nov 5, 2017 at 14:57
• Nov 5, 2017 at 14:58

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 $|A-B|=14$.

• 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. Nov 5, 2017 at 15:10
• $|A\cup B|=28$. 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? Nov 5, 2017 at 15:11
• Thanks, got it now ;) Nov 5, 2017 at 15:20
• We have $2|A|=3|B|=6|A\cap B|=6x$. Since $2|A|=6x$, $|A|=3x$, and since $3|B|=6x$, $|B|=2x$. Nov 5, 2017 at 18:56
• You are great! I got it now. Can you update your answer? It would be good if you can add this into your answer. Nov 5, 2017 at 18:56

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.

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.