# Correct definition of composite relations?

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?

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.
• 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. – GEdgar Feb 23 '19 at 14:27