Cayley table

A truth table is a table, double-digit shortcuts are presented with in mathematics and especially in algebra. For example, the following truth table shows the multiplication on the set:

Link plates occur in the propositional logic in the form of truth tables, for example. In group theory they can be used to write down (small ) group or construct.

Panels of double-digit shortcuts

The presentation as a truth table is suitable for any link. Such a connection assigns to each pair of elements and an element. This assignment can be represented in a table as follows:

In the Input column, the first argument in the header is the second argument, the intersection of the row and column shows the result of the link.

In order to fully write down the table, it is also predicted that the quantities and are finite, and yet sufficiently small for practical purposes.

Frequently linking panels are used for internal links ( ie in the case), and in particular for groups.

Examples

Examples of the logic

Truth tables are used in propositional logic to describe or define the result of logic operations ( connectives ). Three typical examples are

  • The Konjunktor (logical "and"),
  • The Disjunktor (logical "or" ),
  • The implication (logical " if ... then ...").

The following tables show the linkage panels of this connectives:

The first two tables are immediately obvious. The third, however, is less intuitive: you expresses the fact that one can by correct closing of the true conditions only gain true inferences (first line) that can be drawn from false premises but both false and true inferences (second line). This example shows that the logic operations require a clarifying definition, and the truth tables are a suitable notation for this purpose.

Examples from algebra

On the set, we consider two links that addition and multiplication. These correspond to the following two link tables:

Some properties of an internal two -digit shortcut can be read easily from the truth table:

For more examples of linking tables see: Klein's four group, quaternion group, Sedenion, S3 ( group ), A4 ( group).

History

Link panels were first used in group theory by Arthur Cayley. In a work of 1854 he calls them simply panels (English tables ) and uses it to explain groups. In his honor, linking tables are called in group theory and Cayley tables. However, for the construction of linkage groups are panels suitable only for very small groups, as the systematic trial and error greater number of elements is hopelessly inefficient. This approach was therefore supplemented in group theory by more efficient designs and eventually replaced, and plays for the theory are no longer relevant. However, the truth table of a group leads directly to the set of Cayley and therefore a natural starting point for the representation theory of groups.

  • Arthur Cayley, " On the theory of groups, as DEPENDING on the symbolic equation θ n = 1 ," Philosophical Magazine, Vol 7, pp. 40-47. Available online at Google Books as part of his collected works.
  • Arthur Cayley: On the Theory of Groups. In: American Journal of Mathematics, Vol 11, No. 2 ( January 1889 ), pp. 139-157, online available for free at JSTOR 2,369,415th
Propositional calculus Latin square Klein four-group Dihedral group of order 6 Tetrahedral symmetry Cayley's theorem Group representation
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