Are intersection and union redundant? Possibility to get smaller output without negation?
Regarding the first question, as you have said in the comments, you can define a relational algebra intersection with other relational algebra operators. Anyway, relational-algebra union is usually considered a relational algebra basic operator (that cannot be defined with other relational algebra operators).
Regarding monotonicity, we say that a query is monotone if the results of the query increases with the size of the database. More formally, a query $q$ is monotone if and only if, for any given database states $D_1$, $D_2$, where $D_1 \subseteq D_2$, we have $q(D_1) \subseteq q(D_2)$.
For instance, a monotone query might be: &quot;All departments with at least one employee&quot;. Indeed, if you have a database state $D_1$ with 1 department with at least one employee (e.g. 'MarketingDepartment'), you know, for sure, that any $D_2 \supseteq D_1$ will still bring you such 'MarketingDepartment', and possibly more (since $D_2$ might have more departments with at least one employee).
A non-monotone query might be: &quot;All departments with NO employee&quot;. Indeed, if you have a database state $D_1$ with 1 department with no employee (e.g. 'MarketingDepartment'), you can build a database $D_2 \supset D_1$ where 'MarketingDepartment' $\not\in q(D_2)$. Indeed, in $D_2$ you can add some employee for the 'MarketingDepartment', making 'MarketingDepartment' not satisfy your condition.
In relational-algebra, non-monotonic queries are defined through the set-difference operator (in the previous case: Department $\setminus$ $\pi_{dept}$ Employee). If a relational-algebra query has no set-difference operator involved, it is mononote.