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# Def/Transposition

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Definition of Transposition: Suppose that $X$ is a set. A transposition of $X$ is a function $f \colon X \rightarrow X$, such that there exist two distinct elements $a,b \in X$, such that:

• For all $x \in X$, if $x \neq a$ and $x \neq b$, then $f(x) = x$.
• $f(a) = b$.
• $f(b) = a$.

In this case, we say that $f$ transposes, or switches, or exchanges $a$ and $b$.

## Logical Connections

This definition logically relies on the following definitions and statements:

The following statements and definitions logically rely on the material of this page: Def/Sign of a permutation, State/Permutations can be decomposed into transpositions, and State/Uniqueness of prime factorization

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