1. Let S and T be nonempty sets. Prove that | T | ≤ | S | iff there exists a surjection f : S → T 2. Without using the axiom of regularity, show that ∅ ≠ {∅}, and from this conclude that ∅ ≠ ∅ ∪ {∅}....

1. Let S and T be nonempty sets. Prove that | T | ≤ | S | iff there exists a surjection f : S → T

2. Without using the axiom of regularity, show that ∅ ≠ {∅}, and from this conclude that ∅ ≠ ∅ ∪ {∅}.

3. Let x be a set. Show that { y: x ⊆ y} cannot be a set.

4. Use the axiom for regularity to show that for any set x, x ∪ {x} ≠ x