Transformations are rules that convert one data into another data. We may denote a transformation by

` T`:

where ` T` is the transformation
(also called map, function),

A more rigorous theory of transformations can be found here.

Example The following are some transformations in everyday life.

`Age`: people → number

`ID _{s}`: student → number

`ID _{p}`: professor → number

`Instructor`: course → professor

`Maker`: product → manufacturer

`Capital City`: country → city

`Population`: city → number

Example In mathematics, transformations are often given by formulae.

`Square`: **R** → **R**, `x` → `x`^{2}

`Reflection in x-axis`: **R**^{2} →
**R**^{2}, (`x`, `y`) → (`x`, -`y`)

`Polar-Cartesian`: **R**^{2} →
**R**^{2}, (`r`, `θ`) → (`r`cos`θ`, `r`sin`θ`)

`Root-Polynomial`:
**R**^{2} → **R**^{2}, (`x`_{1}, `x`_{2})
→ (`x`_{1} + `x`_{2}, `x`_{1}`x`_{2})

` T`:

Here are two examples of mathematical transformations given by descriptions instead of formulae.

Sign: R → R, |
positive number → 1 |

negative number → -1 | |

0 → 0 |

Dirichlet function: R → R, |
rational number → 1 |

irrational number → 0 |

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