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#1 anlokri  Icon User is offline

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Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 12:01 AM

X = {, {}, {{}}} and R = ⊆ (R is the relation of all ordered pairs where each first co-ordinate is a subset
of the second co-ordinate, and R ⊆ X X.

Please help me in finding all the elements of R in list notation.
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Replies To: Please help me solve the following relation containing empty sets

#2 sepp2k  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 12:07 AM

The set X has 3 elements. So compare each of those elements to each other element and to itself to see whether it's a subset. If element x is a subset of element y, the pair (x,y) will be in R. If not it won't. Once you did this for all 3*3 pairs of elements, you know all the elements in R.
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#3 anlokri  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 03:04 AM

View Postsepp2k, on 11 April 2012 - 12:07 AM, said:

The set X has 3 elements. So compare each of those elements to each other element and to itself to see whether it's a subset. If element x is a subset of element y, the pair (x,y) will be in R. If not it won't. Once you did this for all 3*3 pairs of elements, you know all the elements in R.


Thanks for you help. Would the answer be R = {(,), (,{}), (,{{}}), ({}{}), ({{}},{{}})}? Because 1st coordinate must be subset of 2nd coordinate.

Is it true that ({},{{}}) is not an element of R as shown above. Is {} not a subset of {{}}?

Thanks again for your help.
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#4 sepp2k  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 03:10 AM

Quote

Thanks for you help. Would the answer be R = {(,), (,{}), (,{{}}), ({}{}), ({{}},{{}})}? Because 1st coordinate must be subset of 2nd coordinate.


Yapp, that looks right.

Quote

Is it true that ({},{{}}) is not an element of R as shown above. Is {} not a subset of {{}}?


That's correct. {} contains as an element. {{}} does not contain as an element. So {} can't be subset of {{}} (it is an element of {{}} though, but that doesn't matter for the question).
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#5 anlokri  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 03:57 AM

View Postsepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

Thanks for you help. Would the answer be R = {(,), (,{}), (,{{}}), ({}{}), ({{}},{{}})}? Because 1st coordinate must be subset of 2nd coordinate.


Yapp, that looks right.

Quote

Is it true that ({},{{}}) is not an element of R as shown above. Is {} not a subset of {{}}?


That's correct. {} contains as an element. {{}} does not contain as an element. So {} can't be subset of {{}} (it is an element of {{}} though, but that doesn't matter for the question).


Thank you, appreciate it!
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#6 anlokri  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 06:12 AM

View Postanlokri, on 11 April 2012 - 03:57 AM, said:

View Postsepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

Thanks for you help. Would the answer be R = {(,), (,{}), (,{{}}), ({}{}), ({{}},{{}})}? Because 1st coordinate must be subset of 2nd coordinate.


Yapp, that looks right.

Quote

Is it true that ({},{{}}) is not an element of R as shown above. Is {} not a subset of {{}}?


That's correct. {} contains as an element. {{}} does not contain as an element. So {} can't be subset of {{}} (it is an element of {{}} though, but that doesn't matter for the question).


Thank you, appreciate it!


Just for interest sake. Because {} contains as an element and {{}} does not contain as an element can {{}} be a subset of {}?
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#7 sepp2k  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 06:17 AM

For x to be subset of y, y needs to contain all elements that x contains. In the case of {} and {{}}, both contain an element that the other one does not ({} contains and {{}} contains {}), so neither is a subset or a superset of the other.

Also if you have two sets that have the same number of elements (as you do here), the only way one is a subset of the other is if both sets are equal.
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#8 anlokri  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 06:17 AM

View Postanlokri, on 11 April 2012 - 06:12 AM, said:

View Postanlokri, on 11 April 2012 - 03:57 AM, said:

View Postsepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

Thanks for you help. Would the answer be R = {(,), (,{}), (,{{}}), ({}{}), ({{}},{{}})}? Because 1st coordinate must be subset of 2nd coordinate.


Yapp, that looks right.

Quote

Is it true that ({},{{}}) is not an element of R as shown above. Is {} not a subset of {{}}?


That's correct. {} contains as an element. {{}} does not contain as an element. So {} can't be subset of {{}} (it is an element of {{}} though, but that doesn't matter for the question).


Thank you, appreciate it!


Just for interest sake. Because {} contains as an element and {{}} does not contain as an element can {{}} be a subset of {}?


Will it be correct if I reason that {{}} can NOT be a subset of {} because {{}} would have the element {} which is not found in the set {}?
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#9 sepp2k  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 06:23 AM

Yes, that is correct.
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#10 anlokri  Icon User is offline

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Re: Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 06:24 AM

Thanks again for your help, I now have a much better idea of building subsets from sets containing sets with .
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