Hello,

I'm trying to formalize some properties of tuples and I find some problems for relating the elements of the tuple.

I have the tuple $\displaystyle O$ at time $\displaystyle t$ defined as $\displaystyle O^t=\langle A^t, B^t, C^t, q^t, r^t, s^t\rangle$, where:

- $\displaystyle A^t$ is a set of elements at time $\displaystyle t$

- $\displaystyle B^t$ is a set of elements at time $\displaystyle t$

- $\displaystyle C^t$ is a set of elements at time $\displaystyle t$

- $\displaystyle q^t: A^t \times B^t \rightarrow \{0,1\}$, where $\displaystyle q^t(a,b)=1$ if $\displaystyle a$ and $\displaystyle b$ are related.

- $\displaystyle r^t: A^t \times C^t \rightarrow \{0,1\}$, where $\displaystyle r^t(a,c)=1$ if $\displaystyle a$ and $\displaystyle c$ are related.

- $\displaystyle s^t: B^t \times C^t \rightarrow \{0,1\}$, where $\displaystyle r^t(b,c)=1$ if $\displaystyle b$ and $\displaystyle c$ are related.

Now, I want to represent that: if at time $\displaystyle t$, there is a $\displaystyle q$ relationship that is 1 between $\displaystyle a$ and $\displaystyle b$, and a $\displaystyle r$ relationship that is 1 between $\displaystyle a$ and $\displaystyle c$, then it must exists a relationship $\displaystyle s$ between $\displaystyle b$ and $\displaystyle c$ that is also 1. I don't really know how to write it. I first tried:

$\displaystyle \forall q^t(a,b) \in O^t \wedge r^t(a,c) \in O^t \rightarrow s^t(b,c)$

I think this option is not mathematically correct because $\displaystyle O^t$ is not a set. May the next option be more correct??

$\displaystyle q^t(a,b)=1 \wedge r^t(a,c)=1 \rightarrow s^t(b,c)=1$

I don't know how to say it. I would be very grateful If you could help me.

Thank you.