You have too many braces for y. Did you mean y= {a, d}?
An axiom is always true. Do you mean to ask if the hypotheses of the axiom are true- that is, whether the axiom is applicable here.
Do you mean {a d} ?
Do you mean write the axiom of extensionality with the sentential connectives being conjunctions, disjunctions, and negation only?
The ordinary axiom of extensionality is:
Axy(Az(z in x <-> z in y) -> x = y)
Axy(Az((z in x -> z in y) & (z in y -> z in x)) -> x = y)
Axy(Az((z not in x or z in y) & (z not in y or z in x)) -> x = y)
Axy(Az(not((z not in x or z in y) & (z not in y or z in x))) or x = y)
I guess there's some point to that exercise?
Then as to "the axiom holding true", do you mean, whether, under the axiom, we have {a b} = {a d}. Assuming that a, b, and d are all distinct from one another, then of course {a b} is not equal to {a d}, since b in {a b} and not in {a d} (as well as, d in {a d} and not in {a b}). But we don't need the axiom of extensionality for that, since it follows from idenity theory alone (given, of course, the ordinary definition of {. .} and as enabled by the pairing axiom).