Transposition may refer to:

Logic and Mathematics

Games

Biology

Other uses

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that:

Where "" is a metalogical symbol representing "can be replaced in a proof with."

Formal notation

The transposition rule may be expressed as a sequent:

where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;

or as a rule of inference:

where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "";

or as the statement of a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

In mathematics, and in particular in group theory, a cyclic permutation is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements of X. For example, the permutation of {1, 2, 3, 4} that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a cycle, while the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not (it separately permutes the pairs {1, 3} and {2, 4}).

A cycle in a permutation is a subset of the elements that are permuted in this way. The set S is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into a collection of cycles on disjoint orbits. In some contexts, a cyclic permutation itself is called a cycle.

Definition

A permutation is called a cyclic permutation if and only if it consists of a single nontrivial cycle (a cycle of length > 1).

Example:

Some authors restrict the definition to only those permutations which have precisely one cycle (that is, no fixed points allowed).