# Differences

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 relation_concatenation [2013/09/05 22:33]nikolaj relation_concatenation [2014/03/21 11:11] Line 1: Line 1: - ===== 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]] 