What Are Counterexamples for AεB and BεD Leading to AεD?

[Wildcat]

Homework Statement

Find a counterexample for the following
If AεB and BεD, then AεD.

Homework Equations

The Attempt at a Solution

The answer given is let A={1} B={{1}} and D={{{1}}}
I have not seen this notation before and don't know what it means.
Can someone explain please??

 

[SammyS]

Let's give a different (counter)example: (The highlighting is merely for emphasis.)

Let A={1, 2}

Let B={{1}, {3}, {1, 2}, {1, 2, 4, 8}} ---- You could also write this as: B={{1}, {3}, A, {1, 2, 4, 8}}

Let D={ {{1}, {3}}, {{1, 2, 3}, {1, 2, 4, 8}}, {{1}, {3}, {1, 2}, {1, 2, 4, 8}} }

A is a set whose elements are natural numbers: in this case 1 & 2.

B is a set whose elements are themselves sets: in this case sets of natural numbers, one of which is the set A.

D is a set whose elements are sets of sets. Although the set A is contained in one of the sets of sets, namely set B, which appear in set D, A itself does not appear as one of the elements of set D.

The (counter)example give as the solution is merely a very simple one.

 

[Wildcat]

Let's give a different (counter)example: (The highlighting is merely for emphasis.)

Let A={1, 2}

Let B={{1}, {3}, {1, 2}, {1, 2, 4, 8}} ---- You could also write this as: B={{1}, {3}, A, {1, 2, 4, 8}}

Let D={ {{1}, {3}}, {{1, 2, 3}, {1, 2, 4, 8}}, {{1}, {3}, {1, 2}, {1, 2, 4, 8}} }

A is a set whose elements are natural numbers: in this case 1 & 2.

B is a set whose elements are themselves sets: in this case sets of natural numbers, one of which is the set A.

D is a set whose elements are sets of sets. Although the set A is contained in one of the sets of sets, namely set B, which appear in set D, A itself does not appear as one of the elements of set D.

The (counter)example give as the solution is merely a very simple one.

Your counter example is much easier to understand. I still don't know about the one given. Is it saying in A 1 is a natural number, then B is the set {1} then D ugh I don't know? Can you explain that as simply as your counterexample?

 

[SammyS]

Your counter example is much easier to understand. I still don't know about the one given. Is it saying in A 1 is a natural number, then B is the set {1} then D ugh I don't know? Can you explain that as simply as your counterexample?

They're saying:

A={1}

A is a set whose only element is the number, 1.​

B={ {1} }

B is a set whose only element is the set {1}. You can also say: B is a set whose only element is the set A, because A = {1}
​

D={ { {1} } }

D is a set whose only element is the set B, which is itself a set containing the set A.​

Yes, it can be somewhat confusing. The main idea here is the the number, 1, is not an element of either set B or set D. set B contains a set. That set contains the number 1.

 

[Wildcat]

They're saying:

A={1}

A is a set whose only element is the number, 1.​

B={ {1} }

B is a set whose only element is the set {1}. You can also say: B is a set whose only element is the set A, because A = {1}
​

D={ { {1} } }

D is a set whose only element is the set B, which is itself a set containing the set A.​

Yes, it can be somewhat confusing. The main idea here is the the number, 1, is not an element of either set B or set D. set B contains a set. That set contains the number 1.

Thank you sooo much! I still like your example better :)