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| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
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relation_concatenation [2013/08/06 20:59] nikolaj |
relation_concatenation [2013/09/05 22:33] 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 === | + | |
| Concatenations/compositions are associative. | Concatenations/compositions are associative. | ||
| - | === 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 | ||
| Line 21: | Line 20: | ||
| $(f\circ g)(x):=f(g(x))$ | $(f\circ g)(x):=f(g(x))$ | ||
| + | === 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 ==== | ==== Context ==== | ||
| Set constructor | Set constructor | ||
| === Parents === | === Parents === | ||
| [[Binary relation]] | [[Binary relation]] | ||
