X = {ø, {ø}, {{ø}}} 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.

## 9 Replies - 5877 Views - Last Post: 11 April 2012 - 06:24 AM

### #1

# Please help me solve the following relation containing empty sets

Posted 11 April 2012 - 12:01 AM

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**Replies To:** Please help me solve the following relation containing empty sets

### #2

## 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.

### #3

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

Posted 11 April 2012 - 03:04 AM

sepp2k, 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 = {(ø,ø), (ø,{ø}), (ø,{{ø}}), ({ø}{ø}), ({{ø}},{{ø}})}? Because 1st coordinate must be subset of 2nd coordinate.

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

Thanks again for your help.

### #4

## 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 = {(ø,ø), (ø,{ø}), (ø,{{ø}}), ({ø}{ø}), ({{ø}},{{ø}})}? Because 1st coordinate must be subset of 2nd coordinate.

Yapp, that looks right.

Quote

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

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

### #5

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

Posted 11 April 2012 - 03:57 AM

sepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

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

Yapp, that looks right.

Quote

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

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

Thank you, appreciate it!

### #6

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

Posted 11 April 2012 - 06:12 AM

anlokri, on 11 April 2012 - 03:57 AM, said:

sepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

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

Yapp, that looks right.

Quote

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

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

Thank you, appreciate it!

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

### #7

## 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 {ø} and {{ø}}, both contain an element that the other one does not ({ø} contains ø and {{ø}} contains {ø}), 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.

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.

### #8

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

Posted 11 April 2012 - 06:17 AM

anlokri, on 11 April 2012 - 06:12 AM, said:

anlokri, on 11 April 2012 - 03:57 AM, said:

sepp2k, on 11 April 2012 - 03:10 AM, said:

Quote

Yapp, that looks right.

Quote

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

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

Thank you, appreciate it!

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

Will it be correct if I reason that {{ø}} can NOT be a subset of {ø} because {{ø}} would have the element {ø} which is not found in the set {ø}?

### #9

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

Posted 11 April 2012 - 06:23 AM

Yes, that is correct.

### #10

## 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 ø.

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