Before I begin, note that this is significantly outside what I've studied, so if it's a load of crap just let me know.
Up till now, everything I've covered assumes the same domain for all variables.
I'm trying to define an arbitrary one-to-one relationship between elements in two different sets, and want each of the variables x and y in the formula below to be in the domain of its particular set:
X = {X1, X2, X3}
Y = {Y1, Y2, Y3}
That is, I want:
x ∈ X, y ∈ Y
If possible, how would I go about specifying this?
Also, I've phrased the formula
∀x ∃y [ A(x, y) ∧ ¬A(¬x, y) ]
Where A(x, y) indicates an relationship between x and y. Is this statement well formed? Particularly the ¬x in ¬A(¬x, y)? And does it mean what I think it means - That A(x, y) for all x ∈ X except the x in question, and for the y in question, must be false in order for the statement ot be true? That is, does it define a one-to-one relationship?
Thanks a lot, Wyatt