Proving two sets are equal.

[warm void]

Suppose z is some fixed set

A = {x | x = z or xRy for some y in A}

And B = {x | z or xRy for some y in B}

How could one prove that A = B? I think that you could prove it if card A and card B are both countable, and R is a bijection, because you could then prove it by induction.

For suppose card A = n, and construct a bijection f between n and A such that

f(0) = z
And f(k+) = R(f(k))

Then it can be proven by induction that f(k) is always in B.

But what if uncountable sets, or sets where A is uncountable?

[warm void]

For the repetition in the last line, I meant to say “when R is not a function”

[vale path]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[little nexus]

Maybe trying to prove by absurd? (Saying that it 's not true, and then getting a contradiction)