Is it correct to use a 3-tuple for transitivity in a relation?

[EdgeOfWorld]

correct use of "3-tuple"?

(here I use "A" for the universal quantifier, "E" for the existential quantifier, and "e" to indicate elementhood)

My present definition for Transitivity of a relation:
R is a transitive relation on the set B

AxeB AyeB AzeB [((x,y)eR & (y,z)eR)-->(x,z)eR]

which I shorten to:
Ax,y,zeB[((x,y)eR & (y,z)eR)-->(x,z)eR]


But for a certain proof I need Ey rather than Ay, so I'm wondering if I can use a 3-tuple, (x,y,z), in the following way:

A(x,y,z)eB[((x,y)eR & (y,z)eR)-->(x,z)eR]

or would that mean I'd have to be using B^3??

 

[xxxx0xxxx]

Not sure what you're asking, but transitivity is a property of R, not an operation on the x,y, and z. Thus you can't use this definition to establish the existence of y, or for that matter, x, z, or R itself.

A 3-tuple would be over $$B^3$$

 

[Rasalhague]

Hello, EdgeOfWorld (famous last words, there!), yes, I think your original statement,

$$(\forall x,y,z \in B) [(((x,y)\in R) \& ((y,z) \in R)) \Rightarrow ((x,z) \in R)],$$

is equivalent to

$$(\forall(x,y,z)\in B^3)[(((x,y)\in R) \& ((y,z)\in R))\Rightarrow ((x,z)\in R)].$$

I don't see how this relates to the existence of y, but maybe that's because you haven't told us exactly how you're going to use it.