Problem Set # 1 (due October 2, 2014)
1. Express the following in the language of set theory (without dened symbols).
a. w [(x y) (y x)] [x y].
b. w x y z.
c. = cfw_x.
a. Find P(P(P(P().
b. Express x P( x) in the language of set theo
Problem Set # 4 (due October 23, 2014)
1. Prove that the following are equivalent (i.e. a implies b, and b implies a).
a. There is a one-to-one function f : X.
b. There is a Y X, Y = X and one-to-one function f : X Y .
The above stateme
Problem Set # 5 (due November 7, 2014)
1. For x R (i.e., x is a Dedekind cut), let
x = cfw_r Q : s Q(s x r < s)
|x| = x x
a. Prove x R (i.e., x is a Dedekind cut).
b. Prove x +R x = 0R .
c. Prove: if x R 0R then x = |x|.
Problem Set # 2 (due October 9, 2014)
a. Find < , , , >.
b. Find A, B, C such that (A B) C = A (B C).
c. Prove: (A B) C = (A C) (B C).
2. Let < x, y > = cfw_x, cfw_x, y and < x, y > = cfw_x, cfw_y.
Prove or disprove:
a. < x,
Problem Set # 7 (due November 20, 2014)
a. |< 2| = .
b. |< | = .
c. | R| = |R|.
d. |R R| = |R|.
2. A real number, x, is algebraic if for some n and c0 , . . . , cn Q,
cn xn + cn1 xn1 + + c1 x + c0 = 0.
For example, every ratio
Problem Set # 6 (due November 13, 2014)
R well orders A ON (< A, R > , >)
2. Recall: V0 = , V + = P(V ), and V =
V if LIM .
a. Show for each ON , V is transitive.
b. Show < V V .
c. Show < V V .
3. Dene rank(x) = the leas
Problem Set # 3 (due October 16, 2014)
1. Suppose is an equivalence relation on A and F : A A.
a. Prove [x] = [y] x y.
b. Use a to show distinct elements of A/ are disjoint. This lls in part of
the proof of the equivalence relation/part
Problem Set # 8 (due December 5, 2014)
1. Prove that if is an innite singular cardinal then is the union of < many
sets of cardinality < .
2. Prove that if is a limit ordinal then cf() = if and only if is the smallest
cardinal such that