Monotonicity in data flow analysis I am studying program analysis and synthesis, I was reading a presentation
about some data flow analysis.

I can't seem to understand why $$\bigcap_{s'\in\text{pred (s)}}\text{Out}(s')$$
and $$\operatorname{Gen}(s)\cup(\operatorname{In}(s)\setminus \operatorname{Kill}(s))$$ is monotone, as claimed in
slide 22.

I don't even understand exactly what is the function that is claimed to be monotone.
Can anyone please help ? any help is greatly appreciated! 

*This was cross-posted on Math.SE a couple of days ago
You need to identify the lattices involved in each case. 

Let $(\wp(S),\subseteq)$ be the lattice of all subsets of a set $S$ ordered by subset inclusion. A function $f:\wp(S) \to \wp(S)$ is monotone if for every pair of sets $x$ and $y$, if $x \subseteq y$ it also holds that $f(x) \subseteq f(y)$. The definition applies more generally to functions $f: L \to M$ between partially ordered sets $L$ and $M$. 

Let $Stmt$ be the set of statements occurring in a program and let $(\wp(Stmt),\subseteq)$ be the lattice of subsets of statements ordered by subset inclusion. Let us assume for simplicity that a statement in $Stmt$ is either an assignment $x:= e$ or a Boolean expression $b$ (representing the entry of a conditional or head of a loop). 

A control flow graph has the form $G = (Stmt,E)$ where $E$ is an edge relation over statements. The following definitions are extremely standard:

$succ: \wp(Stmt) \to \wp(Stmt)$ maps a set of statements to their successors.

$succ(X) = \{ y \in Stmt | \text{ there exists } x \text{ in } X \text{ such that } (x,y) \text{ is in } E\}$

$pred: \wp(Stmt) \to \wp(Stmt)$ maps a set of statements to their predecessors.

$pred(X) = \{ y \in Stmt | \text{ there exists } x \text{ in } X \text{ such that } (y,x) \text{ is in } E\}$

It is a standard exercise to show that these are monotone functions. 

Consider the set $Exp$ of all arithmetic and Boolean expressions occurring in a program. The structure $(\wp(Exp),\subseteq)$ is a lattice with the subset inclusion order.

For the available expressions analysis, we can define two functions $Gen$ and $Kill$ in two steps as follows.

First define $Gen: Stmt \to \wp(Exp)$ as a function that maps a statement to the set of expressions generated by that statement. We can define $Gen$ inductively and lift it to sets as follows.

$Gen(x := e) = \{e\}$ and  $Gen(b) = \{b\}$

The lifted function $Gen: \wp(Stmt) \to \wp(Exp)$  is obtained in a standard manner by taking the union of $Gen$ sets for every statement in a given set.

$$ Gen(S) = \bigcup_{s \in S} Gen(s) $$

You can now ask whether this function is monotone.

A note about the slides

The slides you are studying do not contain sufficient information to deduce everything I have said here from first principles and I doubt they are meant to be read that way. I would consult the lecture notes or the section on data flow analysis in a  standard compilers textbook for a thorough treatment with complete definitions.