Consider the relations $r_1, r_2, \ldots , r_n$ and consider a relational expression language as follows:
$r_i$ is a relational expression, $1 \le i \le n$
$e_1 \circ e_2$ is a relational expression if $e_1$ and $e_2$ were relational expressions and where $\circ$ is the composition operator.
$e_1 \cup e_2$ is a relational expression if $e_1$ and $e_2$ were relational expressions and where $\cup$ is the union operator.
For example, $(r_1 \circ r_2) \cup r_2$ is a relational expression. Show that any relational expression $e(r_1, r_2, \ldots , r_n)$ where each $r_i$ appears exactly once is monotonic that is for all $i$, $1 \le i \le n$,
$r_i \subseteq r_i' \rightarrow e(r_1, r_2, \ldots , r_i, \ldots , r_n) \subseteq e(r_1, r_2, \ldots, r_i', \ldots , r_n)$