MA3205 Set Theory AY 2003/2004 Sem 1 NATIONAL UNIVERSITY OF SINGAPORE MATHEMATICS SOCIETY PAST YEAR PAPER SOLUTIONS with credits to Teo Wei Hao MA3205 Set Theory AY 2003/2004 Sem 1 SECTION A Question 1 S i.e. there exists γ ∈ α such that β ∈ γ. Since α is transitive, and so β ∈ α. (a) Let β ∈ α, S This give us α ⊆ α. (b) Let us be given that α is not a limit ordinal. Then ∃β ∈ Ord such that S(β) = α, i.e. β ∈ α, and for all γ ∈ α, we have γ ∈ β or γ = β. Since (α, ∈) is S a linearly ordered S set, there does not exists γ ∈ α such that β ∈ γ. Therefore β 6∈ α. This give us α 6= α. S Instead let us be given that α 6=Sα. S From (1a.), we conclude that α 6⊆ α, i.e. there exists β ∈ α such that β 6∈ α. If γ ∈ α, then we have β 6∈ γ, and since (α, ∈) is a linearly ordered set, we have γ ∈ β or γ = β. Thus α ⊆ β ∪ {β}. Instead if γ ∈ β or γ = β, then since β ∈ α, and α is transitive, we get γ ∈ α. Thus β ∪ {β} ⊆ α. Therefore α = β ∪ {β} = S(β), i.e. α is not a limit ordinal. (c) Let β ∈ P(α). Since α is transitive, for all γ ∈ β, we get γ ∈ α, and from that we get γ ⊆ α. Thus γ ∈ P(α), and we conclude that P(α) is transitive. Question 2 (a) Assume on the contrary that there exists α ∈ Ord such that α ∪ {α} = S(α) = ω. Then since ω is infinite and {α} is finite, we have |ω| = |α ∪ {α}| = |α|. This give us α to be infinite, contradicting the fact that ω is the first infinite ordinal. Thus ω is a limit ordinal. (b) By Axiom of Foundation, there exists β ∈ ω such that for all γ ∈ ω, we have γ 6∈ β. Since ω is transitive, we have β = ∅, and so ∅ ∈ ω. Let α ∈ ω. Since ω is a limit ordinal, we have S(α) 6= ω. Also, we cannot have ω 6∈ S(α), or else it give us ω ∈ α or ω = α, a contradiction to the Axiom of Foundation. Since ordinals are comparable, we conclude that S(α) ∈ ω. Thus ω is an inductive set. (c) Assume on the contrary that ω 6⊆ A. Then X = ω − A is non-empty. Since (ω, ∈) is a well-ordered set, there exists α ∈ X which is the least element of X. As A is inductive, ∅ ∈ A and so α 6= ∅. Since all finite ordinals except ∅ are successor ordinals, there exists β ∈ ω such that S(β) = α. By our condition on α, we have β ∈ A. However A is inductive, which give us α = S(β) ∈ A, a contradiction that α ∈ X. Therefore ω ⊆ A. NUS Math LaTeXify Proj Team Page: 1 of 4 NUS Mathematics Society MA3205 Set Theory AY 2003/2004 Sem 1 Question 3 (a) Since A is countable, there exists a bijection g : A → ω. Let f : P(A) → P(ω) be such that f (X) = g[X]. Since g is a bijection, g[X] = g[Y ] implies X = Y , i.e. f is injective. For all Y ⊆ ω, we have f (g −1 [Y ]) = Y , and so f is surjective. Therefore f is bijective, i.e. |P(ω)| = |P(A)|. (b) Since Q is countable, and R is uncountable, we have R − Q to be uncountable. (c) Let f : (0, 1) → (1, 17) be such that f (x) = 16x + 1. Since f is linear, it is injective. Also for all y−1 y ∈ (1, 17), we have y−1 16 ∈ (0, 1) with f ( 16 ) = y, and so f is surjective. Therefore f is bijective, i.e. |(0, 1)| = |(1, 17)|. Let g : [0, 1] → (0, 1) be a well-defined injective function such that 1 , x = 0; 2 + x x 1 g(x) = , x = , k ∈ Z+ ; k 1 + 2x x, otherwise. y y If y = k1 for some k ∈ N − {0, 1, 2}, then 1−2y ∈ [0, 1] such that g( 1−2y ) = y. 1 If y = 2 , then g(0) = y. Otherwise, g(y) = y. Thus g is surjective. Therefore g is bijective, i.e. |(0, 1)| = |[0, 1]|. Question 4 (a) If |A| < |B| and |B| ≤ |C|, then there exists injective functions f : A → B and g : B → C. This give us gf : A → C to be injective, i.e. |A| ≤ |C|. Assume on the contrary that there exists an injective function h : C → A. This give us hg : B → A to be an injective function, a contradiction to |A| < |B|. Therefore |A| < |C|. (b) If |A| ≤ |B| and |B| < |C|, then there exists injective functions f : A → B and g : B → C. This give us gf : A → C to be injective, i.e. |A| ≤ |C|. Assume on the contrary that there exists an injective function h : C → A. This give us f h : C → B to be an injective function, a contradiction to |B| < |C|. Therefore |A| < |C|. (c) Let f : A × B → B × A be such that f (a, b) = (b, a). f is bijective, and so |A × B| = |B × A|. SECTION B Question 5 (a) If ω1 6∈ Card, then there exists α ∈ ω1 such that α ∈ Card and α = |ω1 |. However, this would implies that α is uncountable, a contradiction to the condition that ω1 is the first uncountable ordinal. NUS Math LaTeXify Proj Team Page: 2 of 4 NUS Mathematics Society MA3205 Set Theory AY 2003/2004 Sem 1 (b) Let f : ω1 → S be such that f (α) = α + 1. By definition of S, f is a bijection, and so |S| = |ω1 |. Since T ⊆ ω1 , |T | ≤ |ω1 |. For all α ∈ Ord, we have ω · α 6∈ S, since it is a limit ordinal. Now if α ∈ ω1 , then α is countable. Thus ω · α is a countable ordinal (since |ω · α| = |ω × α| = ω), and so ω · α ∈ ω1 . Therefore we have the well-defined function g : ω1 → T such that g(α) = ω · α. Let α, β ∈ ω1 such that WLOG, α < β. Then there exists γ ∈ Ord such that α + γ = β. This give us ω · α < ω · α + ω · γ = ω · (α + γ) = ω · β, and so g is injective. Thus |ω1 | ≤ |T |. Therefore by Cantor-Bernstein Theorem, |ω1 | = |T |. (c) Since we have |ω| < |P(ω)|, P(ω) is uncountable. By consequence of Axiom of Choice, there exists uncountable α ∈ Card such that α = |P(ω)|. Since ω1 is the first uncountable cardinal, |ω1 | = ω1 ≤ α = |P(ω)|. Question 6 Let us be given that there exists κ ∈ Card such that |κ| = |R|. Then this give us a bijection f : R → κ. Let us define the relation @ on R, such that for r1 , r2 ∈ R, we have r1 @ r2 iff f (r1 ) ∈ f (r2 ). This resulted in f being the isomorphism between (R, @) and (κ, ∈). Thus (R, @) is a well-ordered set, i.e. R is well orderable. Instead let us be given that R is well orderable, i.e.S(R, @) is a well-order. Let define g : R → V recursively, such that g(x) = {S(g(y)) | y @ x}. Notice that for all x, y ∈ R, if y @ x, then g(y) ∈ g(x), thus g is a order-preserving function. This implies that g is injective, and so (g[R], ∈) is isomorphic to (R, @). Let α ∈ g[R], then there exists x ∈ R such that g(x) = α. Let β ∈ α, i.e. β ∈ S(g(y)) for some y @ x. Then the set X = {y ∈ R | y @ x ∧ β ∈ S(g(y))} is non-empty. Since (R, @) is a well-ordered set, there exists b ∈ X such that b is the least element of X. Assume on the contrary that β ∈ g(b). Then β ∈ g(c) for some c @ b, a contradiction to the condition on b. Therefore from β ∈ S(g(b)) = g(b) ∪ {g(b)}, we conclude that g(b) = β, i.e. β ∈ g[R]. Axiom of Replacement give us g[R] to be a set, and so we can conclude that it is transitive. Thus g[R] ∈ Ord. By definition of cardinals, there exists κ ∈ Card such that κ = |g[R]|. Therefore we have |κ| = |R|. Question 7 (a) Since A is countable, there exists an bijection between A and N. This bijection give us an enumeration of A, i.e. we can write A = {a0 , a1 , a2 , . . .}. Now we define a function f : A → Q ∩ (0, 1) recursively, such that, f (ak ) = sup({0} ∪ {f (ai ) | ai <A ak , i ∈ k}) + inf({1} ∪ {f (ai ) | ai >A ak , i ∈ k}) , 2 k ∈ N. Notice that in this definition above, 0 < ak < 1 for any k ∈ N, also f (ak ) > f (ai ) for any ak >A ai , and f (ak ) < f (ai ) for any ak <A ai , i ∈ k. Thus f is a well-defined, order-preserving function. Since <A is a linear order, we have f to be injective. So by letting B = f [A] ⊆ Q ∩ (0, 1), and g : A → B such that g(a) = f (a) for all a ∈ A, we get g to be an isomorphism between (A, <A ) and (B, <), i.e. (A, <A ) ∼ = (B, <). NUS Math LaTeXify Proj Team Page: 3 of 4 NUS Mathematics Society MA3205 Set Theory AY 2003/2004 Sem 1 (b) Since α < ω1 , we have α to be countable, and so (α, ∈α ) is a countable linearly ordered set. Thus using result of (7a.), we get a set Aα ∈ Q ∩ (0, 1) such that (Aα , <) ∼ = (α, ∈α ). Let f : Aα → α be an isomorphism, and let ∅ 6= X ⊆ Aα . Then since α is well-ordered, and ∅ = 6 f [X] ⊆ α, there exists a least element of f [X], say z. This implies that there exists y ∈ X such that f (y) = z. Since f is an isomorphism, y is the least element of X, and so Aα is well-ordered by <. NUS Math LaTeXify Proj Team Page: 4 of 4 NUS Mathematics Society