MATH 2090  Problem Set #2 (On Sets)

York University

Faculties of Pure and Applied Science, Arts, Atkinson MATH 2090. Problem Set #2. Posted October 8, 2003

Due in the Course Box: October 17, 2003

Section A

In your proofs it is imperative to clearly state what tools you use (e.g., WLUS, s WLUS, MP, Auxiliary Variable, Monotonicity, Deduction Theorem, Generalisation, which axiom(s), etc.)

1. When doing problems from GS, please \underline{convert GS-assertions} to “standard notation” \underline{before} you start your proof.

Conventions AND notation from class apply!

2. Note that we only have the set theory axioms presented in the web documents “Notes on a (very) Elementary Set Theory” Parts I and II.

3. Informal proofs are allowed unless otherwise stated. Like formal proofs, they must be complete and correct.

Marks will be deducted from very long unreadable proofs even if they are correct. Please \underline{think} before you start “proving”.

Probl. 1. From the text p.213–215: Do the problems

11.7(c), 11.12((a), (b), (d)), 11.13(e), 11.15, 11.18

For the first and last in the list above, formal proofs are required.

Also do

Probl. 2. Prove informally ST ⊢ (∀a, b, c, d)({a, {a, b}} = {c, {c, d}} ⇒ a = c ∧ b = d).

MATH 2090  Problem Set #2 (On Sets)

To avoid an embarrassing situation I note that the above is not the same problem that I assigned last year. Do you see the difference?

Probl. 3. Give a formal proof of ST ⊢ A ⊂ B ⇒ (∃x)(x ∉ A).

Probl. 4. Prove \underline{without using the axiom of foundation} that 1 ≠ \{1\} and ∅ ≠ {∅} .