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relation_concatenation [2013/08/06 20:59] nikolaj |
relation_concatenation [2014/03/21 11:11] (current) |
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| ===== Relation concatenation ===== | ===== Relation concatenation ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ R \in \text{Rel}(X,U) $ | | + | | @#55CCEE: context | @#55CCEE: $ R \in \text{Rel}(X,U) $ | |
| - | | @#88DDEE: $ S \in \text{Rel}(V,Y) $ | | + | | @#55CCEE: context | @#55CCEE: $ S \in \text{Rel}(V,Y) $ | |
| - | | @#55EE55: $ \langle x,y \rangle \in S\circ R $ | | + | | @#FFBB00: definiendum | @#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: postulate | @#55EE55: $ \exists m.\ \langle x,m \rangle \in R \land \langle m,y \rangle \in S $ | |
| + | |||
| + | ==== Discussion ==== | ||
| + | Concatenations/compositions are associative. | ||
| - | ==== Ramifications ==== | ||
| - | === 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 | ||
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| then | then | ||
| - | $(f\circ g):X\to Z$ | + | $f\circ g:X\to Z$ |
| $(f\circ g)(x):=f(g(x))$ | $(f\circ g)(x):=f(g(x))$ | ||
| - | + | === Notation === | |
| - | Concatenations/compositions are associative. | + | 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 ==== | + | ==== Parents ==== |
| - | Set constructor | + | === Context === |
| - | === Parents === | + | |
| [[Binary relation]] | [[Binary relation]] | ||
