I want to notate that a predicate holds for all elements in a set. Currently I have the following: $\forall (a,c) \in R^{k+1}(\exists b \in A((a,b) \in R \land (b,c) \in R))$ I want to say that for all ordered pairs (a,c) in $R^{k+1}$ the following applies: There exists an element b in A such that: the order pairs (a,b) and (b,c) are an element of R. Is this notation right or am I doing something wrong?
Edit: Thanks for the answers. I will change the parenthesis. This is indeed unclear. R denotes a relation, a set of ordered pairs. $R^{k+1}$ also denotes a relation. But it has different elements from R. Would switching from a, b and c to x, y and z make it easier to read? Thanks very much in advance
Your symbolization is fine ... though as amWhy comments, usually in logic we use for variables things like $x$, $y$, and $z$, as $a$, $b$, and $c$ are usually used to denote specific objects. So:
$\forall (x,z) \in R^{k+1} \: \Big( \exists y \in A \: \Big( (x,y) \in R \land (y,z) \in R \Big) \Big)$
EDIT
OK, now that I understand that you want $R^k$ to mean all paths of length $k$, you should recursively define $R^{k+1}$ as:
$\forall (x,z) \in R^{k+1} \: \Big( \exists y \in A \: \Big( (x,y) \in R^k \land (y,z) \in R \Big) \Big)$
In fact, this sentence says that only the paths that can be created as such are in $R^{k+1}$ ... not all and only such paths (e.g the statement would be true if $R^{k+1} =\{\}$. So, what you really want is something like:
$\forall x,z \in A \Big( (x,z) \in R^{k+1} \leftrightarrow \exists y \in A \Big( (x,y)\in R^k \land (y,z) \in R \Big) \Big)$
That is, there is a path of length $k+1$ from $x$ to $z$ if and only if there is a path of length $k$ from $x$ to $y$, and one more step from $y$ to $z$.
If you want to prove that if $R$ is transitive then for any $k$ all pairs in $R^{k}$ are in $R$, you want to prove that for all $k$:
$\forall (x,y) \in R^k : (x,y) \in R$
If you want to prove this using formal logic (i.e. Using a formal logic derivation), however, you will need to explicitly quantify over the $k$ as well though, so in both your definition as well as your goal you will have to add $\forall k$ at the beginning (or, if $k$ cannot be used as a variable, use a proper variable). And then you will also need some formalization of induction, where that formalization can deal with the $k$ in $R^k$ ... So maybe even define a function $f(R,k)$ instead of $R^k$ ... How technical/formal do you need to get?