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I can't seem to wrap my head around composite relations. According to Grimaldi in "Discrete and Combinatorial Mathematics", it is defined as:

If $A, B$ and $C$ are sets with $R_1 \subseteq A \times B$ and $R_2 \subseteq B \times C$, then the composite relation $R_1 \circ R_2$ is a relation from $A$ to $C$ defined by $R_1 \circ R_2 = \{(x,z)\mid x\in A, z\in C\}$, and there exists $y\in B$ with $(x,y)\in R_1, (y,z)\in R_2$.

However, in Rosen's "Discrete Mathematics and its Applications", composite relations are defined as:

Let $R$ be a relation from a set $A$ to a set $B$ and $S$ a relation from $B$ to a set $C$. The composite of $R$ and $S$ is the relation consisting of ordered pairs $(a,c)$, where $a\in A, c\in C$, and for which there exists an element $b\in B$ such that $(a,b)\in R$ and $(b,c)\in S$. We denote composite of $R$ and $S$ by $S \circ R$.

Are these two definitions not contradicting?

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Yes, they have different order. If we take first definition, then notation in second definition should be $R\circ S$ and not $S\circ R$. If you are reading first book then stick to definition in that book and if you reading second one then stick definition in that book.

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  • $\begingroup$ In fact you even occasionally see composition of functions in the opposite order. Writing $x^\sigma$ for function $\sigma$ applied to element $x$, we want to define the composition $\sigma\circ\tau$ so that $x^{\sigma\circ\tau} = (x^\sigma)^\tau$. You may see this in some branches of algebra. $\endgroup$ – GEdgar Feb 23 '19 at 14:27
  • $\begingroup$ So this was no help to you? $\endgroup$ – Aqua Jun 14 at 13:53

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