I have the following question:
Relations $S$ and $R$ are defined on the set $$\{1, 2, 3, \ldots, 12\}$$
as follows: $$R = \{(x, y)\mid xy = 12\}$$ $$S = \{(x, y)\mid 2x = 3y\}$$
Write the ordered pairs belonging to $R, \;S,\;$ and $R\circ S$.
I have resolved sets R and S to contain the following:
$$R = \{(1,12),(2,6),(3,4),(4,3),(6,2),(12,1)\}$$ $$S = \{(3,2),(6,4),(9,6), (12, 8\}$$
However I am unsure how to list these two sets as a 'composite relation', can anyone help?
We have $\def\R{\mathrel R}\def\S{\mathrel S}$ $$ R \circ S = \{(x,z) \mid \exists y. \, x\S y\land y \R z\} $$ hence $$ R \circ S = \{(3,6), (6,3), (9,2)\} $$