Finding Disjoint Partitions of a Set: A Problem Solved

[Pengwuino]

I was given a problem where I was to find two disjoint partitions, S_1 and S_2 and a set A such that |A| = 4 and |S_1| = 3 and |S_2| = 3.

Now the set I was using and the book eventually used was A = {1,2,3,4} and S_1 ={{1},{2},{3,4}} and S_2 ={{1,2},{3},{4}}.

The question I have is probably a few definition questions that the book just doesn't seem to be clear about. Do the S's have to be a collection of sets and not simply a set of numbers? For example, is S_1 = {1,2,3} not a correct partition?

Also, the text asks for "disjoint" partitions, which I assume means S_1 and S_2 don't share any elements. However, isn't this part of the definition of a partition? That is, any two sets don't share any elements?

 

[SigmaKID]

A partition of a set A is a collection that simply separates every element of the set A into disjoint non-empty subsets. Every element of the set A must appear once and only once in one of the subsets of the partition. S_1 = {{1,2,3}} is not a partition of the set A since 4 does not appear is any of the subsets. However, S_1 = {{1,2,3}, {4}} would be a partition, however its cardinality would be 2.

It is true that a partition is always a collection of disjoint subsets, however it is possible to have two partitions that are not disjoint, but distinct. For example S_1={{1,2}, {3}, {4}} and S_2={{1,3}, {2}, {4}} would be two distinct partitions of A, both with cardinality 3, however they are not disjoint because they share the element (of a partition which is a subset) {4}.

 

[Pengwuino]

Ok, I think my text is a bit confusing as it made it seem like S1 and S2 were required to make the set A.

For example, would S = {{1,2,3,4}} be a partition of A? And would S={1,2,3,4} be a partition of A?

 

[VeeEight]

A partition must be a set - the first one is correct.
(see definition of partition: http://en.wikipedia.org/wiki/Partition_of_a_set#Definition)