# The injectivity of a function defined on a set of partitions

• Oct 14th 2012, 06:11 AM
Mayk
The injectivity of a function defined on a set of partitions
Hello.

My name is Michael, I'm a freshman CS student and I'm stuck with this problem. I could really use your help:

Let T be a non empty set, and let A and B two sets belonging to P(T), the set of partitions of T ( whatever x belongs to P(T), the complement of x = T - x )

Now, let f be a function, f defined on P(T) with values in the direct product P(A) x P(B), f(x) = ( x intersected with A, x intersected with B )

I must prove that if f is Injective if and only if the Union of A and B is T.

There are two more points to the problem, but I'm hoping if I understand how to do this, I can solve the other two myself. I tried using the Cantor-Bernstein-Schroeder theorem, but I didn't got me anywhere. I could use any help.

Many thanks,

Michael
• Oct 14th 2012, 08:58 AM
Plato
Re: The injectivity of a function defined on a set of partitions
Quote:

Originally Posted by Mayk
Let T be a non empty set, and let A and B two sets belonging to P(T), the set of partitions of T ( whatever x belongs to P(T), the complement of x = T - x )
Now, let f be a function, f defined on P(T) with values in the direct product P(A) x P(B), f(x) = ( x intersected with A, x intersected with B )
I must prove that if f is Injective if and only if the Union of A and B is T.

I quite sure that you clearly understand what you posted. I do not.
It seems T is being used in several different ways.
What exactly does P(T) mean? What are the elements P(T)?
The phrase "the set of partitions of T" has a very well defined meaning.
But it does not seem this question is using the standard definition.

Can you post an example?
• Oct 14th 2012, 01:30 PM
Mayk
Re: The injectivity of a function defined on a set of partitions
Thank you for taking interest in my problem.

To exemplify: Say T = { 1, 2, 3 }. P(T) would be: { { {1}, {2}, {3} }, { {1, 2}, {3}}, { {1, 3}, {2} }, { {1}, {2, 3} }, { 1, 2, 3} }. If x belongs to P(T), x could be any of the partitions of T. Say x is { 1, 2 }. This would mean the complement of x would be T - x. The complement of x is {3} in this case.

In the problem, the exact elements are not specified, it's meant to be a proof for any case.

I'm sorry for not being very clear. And again, thank you for your help. I really need it.
• Oct 14th 2012, 02:07 PM
Plato
Re: The injectivity of a function defined on a set of partitions
Quote:

Originally Posted by Mayk
Thank you for taking interest in my problem.
To exemplify: Say T = { 1, 2, 3 }. P(T) would be: { { {1}, {2}, {3} }, { {1, 2}, {3}}, { {1, 3}, {2} }, { {1}, {2, 3} }, { 1, 2, 3} }. If x belongs to P(T), x could be any of the partitions of T. Say x is { 1, 2 }. This would mean the complement of x would be T - x. The complement of x is {3} in this case.

Thank you for the clarfication. But there is still come missuse of vocabality.
You say "let A and B two sets belonging to P(T)"
That means that of A and B is a partition of T.
How can $A\cup B=T$? How can the union of two partitions of a set equal the set? Partitions are subsets of the power set of a set.
• Oct 15th 2012, 02:16 AM
Deveno
Re: The injectivity of a function defined on a set of partitions
i believe that the OP is mis-using terminology here, and that he (she?) means the power set of T by P(T), NOT "partition of T".
• Oct 15th 2012, 04:09 AM
Plato
Re: The injectivity of a function defined on a set of partitions
Quote:

Originally Posted by Deveno
i believe that the OP is mis-using terminology here, and that he (she?) means the power set of T by P(T), NOT "partition of T".

That may be but it is not consistent with the example posted, The power set of three elements contains eight sets whereas the third Bell number is five as in her/his example.