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relation_concatenation [2013/08/06 20:59] nikolaj |
relation_concatenation [2013/09/05 22:29] nikolaj |
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| @#88DDEE: $ S \in \text{Rel}(V,Y) $ | | | @#88DDEE: $ S \in \text{Rel}(V,Y) $ | | ||
- | | @#55EE55: $ \langle x,y \rangle \in S\circ R $ | | + | | @#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 $ | | | @#55EE55: $ \exists m.\ \langle x,m \rangle \in R \land \langle m,y \rangle \in S $ | | ||
- | ==== Ramifications ==== | + | ==== Discussion ==== |
=== Satisfies === | === Satisfies === | ||
Concatenations/compositions are associative. | Concatenations/compositions are associative. | ||
+ | === 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 $|f(x)|$. | ||
=== Discussion === | === Discussion === | ||
A mayority of uses of the relation concatenation is when the relation is functional, i.e. one composes functions alla | A mayority of uses of the relation concatenation is when the relation is functional, i.e. one composes functions alla |