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relation_concatenation [2013/09/05 22:33] nikolaj |
relation_concatenation [2014/03/21 11:11] |
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| - | ===== Relation concatenation ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $ R \in \text{Rel}(X,U) $ | | ||
| - | | @#88DDEE: $ S \in \text{Rel}(V,Y) $ | | ||
| - | | @#FFBB00: $ \langle x,y \rangle \in S\circ R $ | | ||
| - | |||
| - | | @#55EE55: $ \exists m.\ \langle x,m \rangle \in R \land \langle m,y \rangle \in S $ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | Concatenations/compositions are associative. | ||
| - | |||
| - | A mayority of uses of the relation concatenation is when the relation is functional, i.e. one composes functions alla | ||
| - | |||
| - | $g:X\to Y,\ \ f:Y\to Z$ | ||
| - | |||
| - | then | ||
| - | |||
| - | $f\circ g:X\to Z$ | ||
| - | |||
| - | $(f\circ g)(x):=f(g(x))$ | ||
| - | === Notation === | ||
| - | If $f:X\to Y$ and $g:Y\to Z$ are functions, we'll often denote $f\circ g$ by $fg$. This convenient notation will also be used in more elaborate cases. For example, if by $f(x)$ we done the values of a function $f:X\to\mathbb R$ and $|\cdot|$ is the function which takes a real to its absolute value, then $|f|$ will denote the name of the function with values $|f(x)|$. | ||
| - | ==== Context ==== | ||
| - | Set constructor | ||
| - | === Parents === | ||
| - | [[Binary relation]] | ||
