1. ## Partitioning of Sets

Hey guys. I have this problem:

Let $A = \left\{1,2,...,12\right\}$. Give an example of a partition S of A satisfying the following requirements:
(i) |S| = 5,
(ii) there is a subset T of S such that |T| = 4 and $|\cup_{X\in T} X|$ = 10 and
(iii) there is no element $B \in S$ such that |B| = 3.

I need help understanding what the second part of (ii) means. I was able to create a partition that I think satisfies the other conditions. I just don't see how $|\cup_{X\in T} X|$ could have a cardinality of 10 when the subset T has only 4 elements.

$S=\left\{\left\{1,2,3,4\right\},\left\{5,6\right\} ,\left\{7,8\right\},\left\{9,10\right\},\left\{11, 12\right\} \right\}$

subset $T=\left\{1,2,3,4\right\}$

2. ## Re: Partitioning of Sets

Hi,
What are the elements of a partition S? Why, they are subsets of A. So if T is to be a subset of S, elements of T are subsets of A. Your T does not consist of subsets of A; e.g. 1 is an element of T, but 1 is certainly not an element of S.

You've almost solved your problem. Take T to be a 4 element subset of S with the union of the subsets of A in T totaling 10 integers.

3. ## Re: Partitioning of Sets

Originally Posted by johng
Hi,
What are the elements of a partition S? Why, they are subsets of A. So if T is to be a subset of S, elements of T are subsets of A. Your T does not consist of subsets of A; e.g. 1 is an element of T, but 1 is certainly not an element of S.

You've almost solved your problem. Take T to be a 4 element subset of S with the union of the subsets of A in T totaling 10 integers.
I'm understanding it little by little. The elements of a partition S are $\left\{1,2,3,4\right\},\left\{5,6\right\}, \left\{7,8\right\}, \left\{9,10\right\}, \left\{11,12\right\}$
They are subsets because the elements of the subsets are in A but not every element of A are in the subsets.
So I had T as subsets of A instead of S. So I could have:

$T=\left\{\left\{1,2,3,4\right\},\left\{5,6\right\} , \left\{7,8\right\}, \left\{9,10\right\}\right\}$ making the cardinality 4.

So $\bigcup_{X\in T}X=\left\{1,2,3,4,5,6,7,8,9,10\right\}$ making the cardinality of the union of the subsets equal to 10, and it now satisfies all of the conditions.