Transitive closure for a graph in Monadic Second Order Logic.

Question

$$TC(R, x,y): \forall X \{ \text{ X is R-closed} \wedge x \in X \implies y \in X \}$$ where $\text{ X is R-closed }$ is defined by : $$\forall u,v (u \in X \wedge R(u,v) \implies v \in X) $$

It comes from http://www.labri.fr/perso/courcell/Conferences/ExpoLATA-2.pdf (fifth page)

I cannot understand it. From my point of view it doesn't work.

1. $\forall X$ means "for every subgraph". I cannot imagine a graph which satisfies a such formula. Just taking not connected vertexes ( it is a subgraph) is contrexample- it doesn't satisfy "X is closed".

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So, if we consider $\phi(x,y) = \forall X (X \text { is closed on R } \wedge x \in X) \implies y \in X$

Now, we can say that $(\mathbb{N}, R, y:e, x:0) \models \phi(x, y)$, where $R$ is interpreted: $R(x,y) \iff x = y +2$, is true iff $e$ is even, right?

Answer 1

Perhaps it will help to write the given definitions using more words and fewer logical symbols: the formula $TC(R, x, y)$ says that $x$ is related to $y$ by the transitive closure of the relation $R$ if every $R$-closed set $X$ that contains $x$ also contains $y$; a set $X$ is defined to be $R$-closed if whenever $u$ is related to $v$ by $R$ and $X$ contains $u$, then also $X$ contains $v$.

As an example, let $R$ be the relation on the integers such that $R(x, y)$ holds iff $y = x + 2.$ The transitive closure of $R$ is the relation $TC(R, x, y)$ which holds iff $x$ and $y$ have the same parity (i.e., they are either both even or both odd). The set $X$ that you need to think about in the given formal definition of $TC(R, x, y)$ will be either the set of even integers or the set of odd integers, depending on the parity of $x$.