Let $R$ by a relation on $\mathbb{R}$ defined as $(x, y) \in R \Longleftrightarrow x = 3y$. Describe a relation $R^{2} = R \circ R$.
I seem to struggle with the concept of composing relations. I know how it works on functions and transformations where for $f: A \longrightarrow B$ and $g: B \longrightarrow C$ is the composite $f \circ g: A \longrightarrow C$ which is going to take us from $A$ to $C$. I just fail to see the where are relations taking us from to.
Take for example $(x,z) \in R^2$, then by the definition of composing relations, there exists a $y \in \mathbb{R}$ such that $(x,y) \in R$ and $(y,z) \in R$.
So for example if you take $(9,1) \in R^2$, what $y \in \mathbb{R}$ can make sure that $(9,y) \in R$ and $(y,1) \in R$? The answer would be $y=3$ as $(9,3) \in R$ and $(3,1) \in R$.
Now, can you find the condition for the values of $x$ and $y$ so that the composing relation always exists?