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 Post subject: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 19:46:35 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
Posts: 141
Hello all

Union operation is a well-known operation used with sets.

Assume A, B, and C are three different sets.

1 - (|B U C|/ |A U B U C|)

How can we simplify the above formula to

|A|-|B| / |A|+|C|

Thanks for any help

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 19:54:39 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15562
Location: Austin, TX
Hello all

Union operation is a well-known operation used with sets.

Assume A, B, and C are three different sets.

1 - (|B U C|/ |A U B U C|)

How can we simplify the above formula to

|A|-|B| / |A|+|C|

Thanks for any help

What is your ||? Cardinality? Measure?

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 20:20:40 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
Posts: 141
it is cardinality or set size

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 20:24:37 UTC
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it is cardinality or set size

Then what happens if some of them have infinite size? Are you assuming finite sets?

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 20:53:13 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
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Yes, I am assuming finite sets..

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 21:57:35 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15562
Location: Austin, TX
Hello all

Union operation is a well-known operation used with sets.

Assume A, B, and C are three different sets.

1 - (|B U C|/ |A U B U C|)

How can we simplify the above formula to

|A|-|B| / |A|+|C|

Thanks for any help

I don't buy it, let A be empty and B= {1}, C={2}, then |BUC|=2, and |AUBUC|=2, so the first equation is 0, but the second is -1/1=-1 which is false. Indeed, so long as |B|<|A| the second one is always negative, but the first one is clearly always positive.

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 Post subject: Re: simplify an equation related to SetsPosted: Fri, 9 Mar 2012 22:24:33 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
Posts: 141
The formulas work with these conditions:
|A| > 0
|B| >= 0
|C| >= 0
and
|B| <= |A|

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 00:53:51 UTC
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The formulas work with these conditions:
|A| > 0
|B| >= 0
|C| >= 0
and
|B| <= |A|

Still not true, let

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 00:58:36 UTC
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And if you think requiring |C|>0 will help, it doesn't. Use the same example and let and it's still false.

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 08:17:31 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: NCTS/TPE, Taiwan
And if you think requiring |C|>0 will help, it doesn't. Use the same example and let and it's still false.

Or something like .

In fact, to make the statement

true, you can choose the cardinalities of ,,,, (more-or-less) arbitrarily, only needing to adjust to make sure is a nonnegative integer.

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 10:36:40 UTC
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outermeasure wrote:
And if you think requiring |C|>0 will help, it doesn't. Use the same example and let and it's still false.

Or something like .

In fact, to make the statement

true, you can choose the cardinalities of ,,,, (more-or-less) arbitrarily, only needing to adjust to make sure is a nonnegative integer.

Exactly, unless there are more data, though, this is quite false, and certainly for arbitrary sets this is very false.

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 10:50:08 UTC
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outermeasure wrote:
And if you think requiring |C|>0 will help, it doesn't. Use the same example and let and it's still false.

Or something like .

In fact, to make the statement

true, you can choose the cardinalities of ,,,, (more-or-less) arbitrarily, only needing to adjust to make sure is a nonnegative integer.

Exactly, unless there are more data, though, this is quite false, and certainly for arbitrary sets this is very false.

Yes.

An easy way to satisfy the conclusion is replacing the condition |B|<=|A| with . But that is of no fun!

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 10:53:39 UTC
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outermeasure wrote:
outermeasure wrote:
And if you think requiring |C|>0 will help, it doesn't. Use the same example and let and it's still false.

Or something like .

In fact, to make the statement

true, you can choose the cardinalities of ,,,, (more-or-less) arbitrarily, only needing to adjust to make sure is a nonnegative integer.

Exactly, unless there are more data, though, this is quite false, and certainly for arbitrary sets this is very false.

Yes.

An easy way to satisfy the conclusion is replacing the condition |B|<=|A| with . But that is of no fun!

YES! In fact that was the only way short of being very specific about the sizes of the piece of the Venn Diagram that actually made it true, and it was utterly trivial for that case.

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 11:25:27 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
Posts: 141
Thank you

Yes..replace |B| <= |A| with

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 Post subject: Re: simplify an equation related to SetsPosted: Sat, 10 Mar 2012 15:01:57 UTC
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Joined: Tue, 7 Aug 2007 11:37:58 UTC
Posts: 141
Sorry brothers..

The last condition is not enough because of

.. 1/3 vs. 1/4 !!

The last updated list of the conditions is:

|A| > 0

|B| >= 0

|C| >= 0

Hope no change later

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