Covering and Partition

Covering

           Let S be a set and let S1,S2,,,,,,,,Sn be the subsets of S such that S1unionS2unionS3.......union Sn=S. Then S1, S2, S3....... Sn are said to be covers of S

Ex let S={1,2,3,4} and
A1={{1,2,3},{4,5}}
A2={{1,2,3},{3,5}}
A3={{1,2,3,4}}
A4={{1,2,3},{3,4,5}}

Then A1 is said to be cover of S
         A2 is not a cover of S since 4 is missing
         A3 is said to be cover of S
         A4 is said to be cover of S


Partition:
          Let S be a set and let S1,S2,,,,,,,,Sn be the disjoint  subsets of S such that                                               S1unionS2unionS3.......union Sn=S. Then S1, S2, S3....... Sn are said to be partitions of S.

Ex let S={1,2,3,4} and
A1={{1,2,3},{4,5}}
A2={{1,2,3},{3,5}}
A3={{1,2,3,4}}
A4={{1,2,3},{3,4,5}}

Then A1 is said to be cover of S
         A2 is not a cover of S since 4 is missing
         A3 is said to be cover of S
        A4 is not a partition since the given two subsets are not disjoint i.e., 3 is the common element.

Note: Every partition is a covering but not all  coverings are partitions



partition and covering, hass diagram, poset, lattice, pigeon hole principle, inclusion and exclusion principle transitive closure, compatibility relation