1. ## Axiomatic Set theory

1. Given the following sets:
x = {a, b}
y= {a,d}}

Write the Axiom of Extensionality in expanded form using conjunctions and
disjunctions.
Does the axiom hold true for this example?

2. 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.

3. Originally Posted by aaronrj
1. Given the following sets:
x = {a, b}
y= {a,d}}

Write the Axiom of Extensionality in expanded form using conjunctions and
disjunctions.
Does the axiom hold true for this example?
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).