Lemma of the shepherds

The lemma of the shepherds is a commonplace property used in Mathématiques, in particular in analyzes combinative.

It can be stated at the elementary level by:

  • If a Ensemble E has a partition in p subsets containing each one R elements, then E contains p × R elements.
For example, a play of Bridge has a partition four colors comprising each one thirteen charts, the full number of charts is thus equal to fifty-two.

One frequently uses this lemma in the other direction:

  • If one knows the number of elements of E, and if E admits a partition in p subsets with R elements (one of the numbers p and R being known but not the other), one from of deduced that from the numbers p and R which one did not know.
The etymology of the nickname of this property comes from the picturesque form of the reciprocal one: When the shepherds want to count their sheep, they count the legs and divide by four.

A version more abstracted from the Théorème is stated as follows:

  • being given two finished units, X and Y, and a surjective application F :   X  →  Y such as any element of Y has exactly N Antécédent S in X, then one a    Card (X)   =  N ×Card (Y)

See too

Random links:Commune of Knivsta | Callabiana | Rossa (Italy) | Wijchen | Leon Crouslé | Liste_des_maladies_(t)