Inclusion Exclusion - XP Math - Forums

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04-27-2007   #1
Sillysidley

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Inclusion Exclusion

Let U=(1,2,3..,1000) and let A2,A3, and A5 be the numbers that are divisible by 2,3, and 5 respectively. Find:
a. The intersection of all 3 sets (I solved this already)
b. The intersection of A2,A3, and the compliment of A5
c. The intersection of A2, the compliment ofA3, and A5.
d. The intersection of the compliment of A2,A3, and A5
e. The intersection of A2, the compliment of A3, and the compliment of A5
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Last edited by Sillysidley; 04-27-2007 at 06:30 PM..

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 04-28-2007 #2 Temperal Guest   Posts: n/a a) To solve this, we need to find all numbers x where 2a=x, 3b=x and 5c=x. These are all primes, so we could try multiplying. 2*3*5=30 4*3*5=60 6*3*5=90 8*3*5=120 We see a pattern here. Since it must end in 0 or 5, it must be even, and divisible by 3, this is easy. The intersection is I=(30,60,90,120,150...) all the way up to 1,000. Is it right, somedude? I'll do the rest later...
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 04-28-2007 #3 Temperal Guest   Posts: n/a BTW, it's spelled "complement", not "compliment".
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04-28-2007   #4
Sillysidley

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Oh well.
All you really need to do is draw a Venn Diagram Don't worry, I did it using the Inclusion Exclusion Property.
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04-28-2007   #5
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Quote:
 Originally Posted by Archive a) To solve this, we need to find all numbers x where 2a=x, 3b=x and 5c=x. These are all primes, so we could try multiplying. 2*3*5=30 4*3*5=60 6*3*5=90 8*3*5=120 We see a pattern here. Since it must end in 0 or 5, it must be even, and divisible by 3, this is easy. The intersection is I=(30,60,90,120,150...) all the way up to 1,000. Is it right, somedude? I'll do the rest later...
You just divide 1000 by 30 and round down.
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04-29-2007   #6
Sillysidley

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Ok I figured them all out, using a Venn Diagram
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 04-29-2007 #7 Temperal Guest   Posts: n/a I hate Venn Diagrams. I just solved the last one NOT using a venn diagram. I wonder if you can make venn diagrams in $\LaTeX$...
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 05-01-2007 #8 Scion Last Achievements   Join Date: Nov 2006 Posts: 446 Thanks: 0 Thanked 674 Times in 343 Posts 5 times 3 times 2 us 30 count by 30 to 1000 __________________
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 05-01-2007 #9 Temperal Guest   Posts: n/a That's been confirmed already. Man, Reds, you wouldn't survive a minute on AoPS.
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05-01-2007   #10
Sillysidley

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For the last time, instead of counting from 30 to 1000, just divide 1000 by 30!
Oh, everyone would yell at him for confirming an already said answer,
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