Give an example of a partition of Z indexed by a finite set

Give an example of a partition of Z indexed by a finite set the set of natural numbers N Make sure to carefully check that the partition you give is actually a partition.

Solution

The set { 1, 2, 3 } has these five partitions (one partition per item): { {1}, {2}, {3} }, sometimes written 1|2|3. { {1, 2}, {3} }, or 12|3. { {1, 3}, {2} }, or 13|2. { {1}, {2, 3} }, or 1|23. { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number). The following are not partitions of { 1, 2, 3 }: { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set. { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block. { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partition of {1, 2}