# Math Help - Formalization of tuples

1. ## Formalization of tuples

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 $O$ at time $t$ defined as $O^t=\langle A^t, B^t, C^t, q^t, r^t, s^t\rangle$, where:
- $A^t$ is a set of elements at time $t$
- $B^t$ is a set of elements at time $t$
- $C^t$ is a set of elements at time $t$
- $q^t: A^t \times B^t \rightarrow \{0,1\}$, where $q^t(a,b)=1$ if $a$ and $b$ are related.
- $r^t: A^t \times C^t \rightarrow \{0,1\}$, where $r^t(a,c)=1$ if $a$ and $c$ are related.
- $s^t: B^t \times C^t \rightarrow \{0,1\}$, where $r^t(b,c)=1$ if $b$ and $c$ are related.

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

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

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

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

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

Thank you.

2. Originally Posted by klendo
$q^t(a,b)=1 \wedge r^t(a,c)=1 \rightarrow s^t(b,c)=1$
This seems correct. You may need to quantify this statement over all a, b, c, and possibly t.

I'd suggest using relations instead of functions q, r, s (they are characteristic functions of relations).