Question

Give an example of set A with |A| = 5 and two disjoint partitions S1and S2ofA with |S1| = |S2|= 3.

Answer 1

`A=\{ 1,2,3,4,5\},~~~S_1=\{ \{1\},\{2\},\{3,4,5\}\}~~~S_2=\{ \{1,2,3\},\{4\},\{5\}\}`

Explanation:

Notice the set
`A` has 5 elements (which is denoted as |A|=5). A partition of A is kind of like distributing the elements of A into as many sets as you want, without leaving any set empty (formally it's just a collection of nonempty disjoint subsets of A whose union is all of A).

In the partition
`S_1` we distribute the elements of A into three sets, resulting with the sets
`\{ \{1\},\{2\},\{3,4,5\}\}` (so basically 1 and 2 are in their own sets, and 3,4,5 are all in another set). This partition is made then of 3 sets (since we distributed the elements of A into this three sets), and so
`|S_1|=3`

In the partition
`S_2` we distribute the elements of A into three sets also, resulting with the sets
`\{ \{1,2,3\},\{4\},\{5\}\}` (so basically 4 and 5 are in their own sets, and 1,2,3 are all in another set). This partition is made then of 3 sets (since we distributed the elements of A into this three sets), and so
`|S_2|=3`

The partitions
`S_1` and
`S_2` are disjoint, since they don't have any element in common. The elements of the partitions are technically the SETS that are inside of them, and notice that both partitions are made of entirely different sets.