Examples and Definitions
The notion of covering system was introduced by Paul Erdős in the early 1930s.
The following are examples of covering systems:
and
and
A covering system is called disjoint (or exact) if no two members overlap.
A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).
A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.
The first two examples are disjoint.
The third example is distinct.
A system (i.e., an unordered multi-set)
of finitely many residue classes is called an -cover if it covers every integer at least times, and an exact -cover if it covers each integer exactly times. It is known that for each there are exact -covers which cannot be written as a union of two covers. For example,
is an exact 2-cover which is not a union of two covers.
Read more about this topic: Covering System
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