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Home , Bijection, Cardinal number

ON THE CONNECTIONS BETWEEN THEORY AND THEORY

DINGDING DONG

Abstract. This paper is an introduction to the connections between set the- ory and . After introducing the notion of cardinal , we prove the Schroeder-Bernstein Theorem about cardinal equivalence. We then discuss how sets and groups exhibit the properties of each other, specifically, the cardinal relations between groups and the group structures within sets of different .

Contents 1. Introduction 1 2. Cardinal Arithmetic 1 3. Schroeder-Bernstein Theorem 4 4. Lagrange’s Theorem 6 5. Cyclic Groups 7 6. Understanding Sets within Group Structures 8 Acknowledgments 10 References 10

1. Introduction The exploration of uncountable sets dates back to the nineteenth century, when proved the existence of infinite sets that cannot be put into one- to-one correspondence with the set of natural . Through the diagonal argument, Cantor showed that there is no between the sets of real and natural numbers. Stemmed from this discovery, the notion of helped mathematicians understand more about sets that seem not as approachable. Groups are a particular kind of sets equipped with a binary , which satisfies the four group of closure, associativity, identity and invertibility. Since groups are still sets, anything about sets still holds true in groups. However, due to their specific properties, there is more accessible information about groups themselves.

2. Cardinal Arithmetic Cardinal numbers measure the cardinality, or size of sets. In terms of bijective functions, it defines an equivalence between sets. Definition 2.1. Two sets A and B are of the same cardinality if there exists a bijection f : A → B between A and B. 1 2 DINGDING DONG

Definition 2.2. A cardinal is thought of as an equivalence of cardi- nality, i.e. card(A) = card(B) if A and B are of the same cardinality. Definition 2.3. Let a, b be cardinal numbers. If there exist sets A, B with card(A) = a, card(B) = b and A ⊆ B, then a ≤ b. Definition 2.4. Let a, b be cardinal numbers. If A, B are sets with card(A) = a, card(B) = b and A ∩ B = ∅, then a + b = card(A ∪ B). Definition 2.5. Let a, b be cardinal numbers. If A, B are sets with card(A) = a and card(B) = b, then a · b = card(A × B).

Definition 2.6. ℵ0 = card(N).

Theorem 2.7. For any infinite a, ℵ0 ≤ a.

Proof. Let A be an infinite set with card(A) = a, and F0 ⊂ F1 ⊂ F2 ⊂ ... be a of of A with |Fi| = i. Let xi ∈ Fi \ Fi−1. Then x1, x2, ... ∈ A with xi 6= xj for any i 6= j, {xi | i ∈ N} ⊆ A, card({xi | i ∈ N}) ≤ card(A), ℵ0 ≤ a.  An important lemma in is Zorn’s Lemma, which identifies the existence of a maximal within any that is bounded above. Definition 2.8. A partially ordered set is a set P with a ≤ such that, for all a, b, c ∈ P :

(1) (Reflexivity) a ≤ a. (2) (Antisymmetry) If a ≤ b and b ≤ a, then a = b. (3) (Transitivity) If a ≤ b and b ≤ c, then a ≤ c. Lemma 2.9. (Zorn’s Lemma) Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element. Theorem 2.10. Let a be an infinite cardinal number. Then a + a = a. Proof. Following the proof of Z. Nagy, let A be an infinite set with card(A) = a. Let S(A) be the set of families {Ai | i ∈ I} of subsets of A with

(a) card(Ai) = ℵ0 for all i ∈ I; (b) Ai ∩ Aj = ∅ for all i, j ∈ I.

S(A) with set inclusion is a partially ordered set. For any S ∈ S(A), S ⊆ S. If S1,S2 ∈ S(A), S1 ⊆ S2 and S2 ⊆ S1, then S1 = S2. If S1,S2,S3 ∈ S(A), S1 ⊆ S2 and S2 ⊆ S3, then S1 ⊆ S3. Since S(A) with set inclusion is a partially ordered set and every chain in S(A) is bounded above by {Ai | Ai ⊆ A, card(Ai) = ℵ0}, S(A) contains a maximal element {Ai | i ∈ I0}. Let B = A\(S A ). Since {A } is maximal, B is finite, card(A) = card(S A ). i∈I0 i i i∈I0 i Then a = card(A) = card(N × I0). Suppose card(I0) = c. Then a = ℵ0 · c. Let C0 = {n ∈ N | n even}, C1 = {n ∈ N | n odd}. Since there are f(x) = 2x between N and C0, g(x) = 2x − 1 between N and C1, card(C0) = card(C1) = ℵ0. Since (C0 × I0) ∪ (C1 × I0) = N × I0,(C0 × I0) ∩ (C1 × I0) = ∅, ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY 3

a = card(N × I0) = card(C0 × I0) + card(C1 × I0)

= ℵ0 · card(I0) + ℵ0 · card(I0) = ℵ0 · c + ℵ0 · c = a + a.



Theorem 2.11. If a, b are cardinal numbers with 1 ≤ b ≤ a and a infinite, then a · b = a.

Proof. Since a ≤ a · b ≤ a · a, it is sufficient to prove that a = a · a. Define χ = {(D, f): D ⊂ A, f : D → D × D bijective}, and equip χ with the

( 0 0 0 D ⊂ D (D, f) ≺ (D , f ) ⇐⇒ 0 . f = f |D S Let τ = {(Di, fi): i ∈ I} be a totally ordered of χ, D0 = i∈I Di, f : D0 → D0 × D0 be the unique with f|Di = fi. Then f is injective with [ [ [ f(D0) = f(Di) = fi(Di) = (Di × Di) = D0 × D0. i∈I i∈I i∈I (D, f) is an upper bound of τ. According to Zorn’s Lemma, since ∀τ ⊆ χ is bounded above, there exists a maximal element (D, f) ∈ χ. Let card(D) = d. Since (D, f) ∈ χ, d · d = d. If d = a, then a · a = a. By contradiction, assume that d 6= a. Since d ≤ a as D ⊂ A, d < a. Let G = A \ D. Then d + card(G) = a. Since d < a, card(G) = a by Theorem 2.10, and there exists a subset E ⊂ G with card(E) = d. Consider the set

P = (E × E) ∪ (E × D) ∪ (D × E).

Since E ∩D = ∅,(E ×E), (E ×D), (D×E) are disjoint from each other. According to Theorem 2.10,

card(P ) = card(E × E) + card(E × D) + card(D × E) = d · d + d · d + d · d = d = card(E).

There exists a bijection g : E → P . Since E ∩ D = P ∩ (D × D) = ∅, there exists a bijection h : D ∪ E → P ∪ (D × D) such that h|D = f and h|E = g. Since P ∪(D ×D) = (D ∪E)×(D ∪E), (D ∪E, h) ∈ χ,(D, f) is not the maximal element in χ. ⇒⇐ Hence d = a, a · a = a.  4 DINGDING DONG

3. Schroeder-Bernstein Theorem Schroeder-Bernstein Theorem states that if there exists an injection from set A to set B and an injection from set B to set A, then there exists a bijection between A and B. In terms of cardinality, it means that if |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|, which completes the ordering of cardinal numbers. Proposition 3.1. If B ⊆ A and there exists an f : A → B, then there exists a bijective function h : A → B between A and B. Proof. If X ⊆ Y , then {f(k) | k ∈ X} ⊆ {f(k) | k ∈ Y }, f(X) ⊆ f(Y ). Since in the base case, we have A ⊇ B, B ⊇ f(A), A ⊇ B ⊇ f(A) ⊇ f(B) ⊇ f 2(A) ⊇ f 2(B) ⊇ ... by induction. Consider the sequence of sets

X0 = A \ B,X1 = B \ f(A),X2 = f(A) \ f(B), ...

Since for any n, Xn+2 = f(P ) \ f(Q) while Xn = P \ Q, for any x ∈ Xn+2, x ∈ f(P ), there exists p ∈ P such that f(p) = x. Since p∈ / Q (or x = f(p) ∈ f(Q)), p ∈ P \ Q = Xn with f(p) = x. Hence f is both surjective and injective, there exists a bijection f : Xn → Xn+2 between Xn and Xn+2. For any n, m ∈ N such that n > m, since Xm ∩ Xm+1 = ∅ and Xm+1 ⊇ Xn, Xm ∩ Xn = ∅. Hence for sets ∞ ∞ [ [ V = Xn,W = Xn, n=0 n=1 there exists a bijection F : V → W between V and W with ( x if x ∈ X , n odd F (x) = n . f(x) if x ∈ Xn, n even Since B ⊆ A, V ⊆ A, W ⊆ V , W ⊆ B, A = (A \ (B ∪ V )) ∪ (B \ V ) ∪ (V \ B) ∪ ((B ∩ V ) \ W ) ∪ W B = (B \ V ) ∪ ((B ∩ V ) \ W ) ∪ W V = (V \ B) ∪ ((B ∩ V ) \ W ) ∪ W with (A \ (B ∪ V )), (B \ V ), (V \ B), ((B ∩ V ) \ W ),W disjoint from each other. Since A \ B = X0 = V \ W , (A \ (B ∪ V )) ∪ (V \ B) = A \ B = V \ W = (V \ B) ∪ ((B ∩ V ) \ W ) (A \ (B ∪ V )) = ((B ∩ V ) \ W ) A \ V = (A \ (B ∪ V )) ∪ (B \ V ) = (B \ V ) ∪ ((B ∩ V ) \ W ) = B \ W Let Z = A \ V = B \ W . Since there exists a bijection f : V → W between V and W , there exists a bijection h : A → B between A and B with ( F (x) if x ∈ V h(x) = . x if x ∈ Z

 ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY 5

Theorem 3.2. (Schroeder-Bernstein Theorem) If there exist injective functions f : A → B and g : B → A, then there exists a bijective function h : A → B between A and B. Proof. Since g : B → A, g(B) ⊆ A. Since there exists an injective function g ◦ f : A → g(B), according to Proposition 3.1, there exists a bijective function h0 : A → g(B) between A and g(B). Since g−1 : g(B) → B is a bijective function between −1 0 g(B) and B, h = g ◦ h : A → B is a bijective function between A and B.  A graph theory argument can also be used to prove Schroeder-Bernstein Theo- rem: Remark 3.3. There is an alternative proof of Theorem 2.2 with graphs. Following the idea of L. Babai, consider the bipartite graph G between A and B that connects a to f(a) for all a ∈ A, and b to f(b) for all b ∈ B. Then for any x ∈ A ∪ B, deg(x) is either 1 or 2. S Let {Vi : i ∈ I} be the set of connected components of G. Then Vi = G. P P i∈I Let Ai = Vi ∩ A, Bi = Vi ∩ A. Then |Ai| = |A|, |Bi| = |B|. For any Vi, Vi is a connected graph with each vertex having deg(v) = 1 or 2. If Vi contains a cycle, then each vertex in the cycle cannot connect with any more vertices, Vi is a cycle. If Vi does not contain a cycle, then there cannot be more than one path within Vi, Vi is a path. If Vi is not a two-side infinite path, then there is an endpoint of Vi with either e ∈ A or e ∈ B, Vi is ei- ther the path connecting {e, f(e), g(f(e)), f(g(f(e))), ...} or the path connecting {e, g(e), f(g(e)), g(f(g(e))), ...}. Hence Vi is either a cycle, a two-side infinite path, a one-side infinite path starting from set A, or a one-side infinite path starting from set B. By picture, Vi is of one of the following forms:

(1) (2)

(3) (4) P P Hence within any Vi, |Ai| = |Bi|. Since |A| = |Ai| = |Bi| = |B|, A, B are of the same cardinality, there is a bijection between A and B. 6 DINGDING DONG

4. Lagrange’s Theorem As a particular kind of sets, groups themselves contain more accessible informa- tion. Lagrange’s Theorem states that for any finite group G, the order (number of elements) of every H of G divides the order of G. Definition 4.1. A group hG, ∗i is a set G closed under a binary operation ∗ such that the following axioms are satisfied:

(1) (Associativity) For all a, b, c ∈ G,(a ∗ b) ∗ c = a ∗ (b ∗ c). (2) (Identity element) There is an element e ∈ G such that for all x ∈ G, e ∗ x = x ∗ e = x. (3) () For all a ∈ G, there is an element a−1 ∈ G such that a ∗ a−1 = a−1 ∗ a = e. Notation 4.2. The operation ∗ is sometimes omitted for convenience, in which case a ∗ b is represented by ab. Definition 4.3. A subgroup H of G is a subset of G closed under the binary operation of G, and is itself a group. H ≤ G and G ≥ H denote that H is a subgroup of G. Definition 4.4. Given H ≤ G, a coset of H in G is gH = {gh | h ∈ H}. Lemma 4.5. For g ∈ G, gH = H if and only if g ∈ H. Proof. For any g∈ / H, since e ∈ H, ge ∈ gH and ge = g∈ / H, gh 6= H. For any g ∈ H, gH = {gh | h ∈ H} ⊆ H as H is a closed subgroup. For any h ∈ H, since h = (gg−1)h = g(g−1h) and g−1h ∈ H, h ∈ gH, H ⊆ gH. Hence gH = H. Hence gH = H if and only if g ∈ H.  Proposition 4.6. For g, g0 ∈ G, gH = g0H if and only if g−1g0 ∈ H. Proof. If gH = g0H, then H = g−1gH = g−1g0H. According to Lemma 4.5, H = −1 0 −1 0 0 −1 0 g g H if and only if g g ∈ H. Hence gH = g H if and only if g g ∈ H.  Proposition 4.7. All cosets of H have the same size as H. Proof. For any coset gH, f : h → gh is a bijective between H and gH. Since for any h ∈ gH, there exists h0 ∈ H such that f(h0) = h, the map is surjective. Since for any h1, h2 ∈ H such that f(h1) = f(h2), there is −1 −1 h1 = g gh1 = g gh2 = h2, the map is injective. Hence there is a bijection between any coset gH and H, all cosets of H has the same size as H. 

Proposition 4.8. For any two cosets of H in G, if g1H ∩ g2H 6= ∅, then g1H = g2H.

Proof. If g1H ∩ g2H 6= ∅, then there exist h1, h2 ∈ H such that g1h1 = g2h2, −1 −1 g1 g2 = h2 h1 ∈ H. According to Proposition 4.6, g1H = g2H.  Proposition 4.9. For any finite group G and its subgroup H, there are totally |G|/|H| cosets of H in G. ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY 7

Proof. Since each g ∈ G lies in the coset gH, the of all cosets S gH = G. According to Proposition 4.8, any two cosets of H in G are either identical or disjoint, so there exists a set of disjoint cosets {gi} such that [ [ giH = gH = G X |giH| = |G|.

According to Proposition 4.7, each coset giH has |giH| = |H|, so |{giH}| = |G|/|H|.  Theorem 4.10. (Lagrange’s Theorem) For any finite group G, the order of every subgroup H of G divides the order of G.

Proof. According to Proposition 4.9, |G|/|H| = |{giH}|. Hence |H| divides |G|.  Corollary 4.11. If g ∈ G, n is the minimal positive with gn = e (i.e. n is the order of g ∈ G), then n | |G|.

5. Cyclic Groups Cyclic group are groups generated by a single element. Despite the multitude of cyclic groups, they can all be summed up as of Zn and Z with identifiable generators.

Definition 5.1. Given a group G and an element a ∈ G. If G = {an | n ∈ Z}, then the group G = hai is cyclic, and a is a generator of G. Definition 5.2. If a cyclic group G = hai is finite, then the order of a is the order |hai| of this cyclic group.

Theorem 5.3. Zn and Z are the only cyclic groups up to . Proof. Let G be a cyclic group generated by a. By cases, if G = {e}, then H is isomorphic to Z0. If G 6= {e} and is finite, then there exists a 6= e ∈ G, and n ∈ N such that an ∈ G. Let m be the smallest positive integer such that am = e. Consider the f : n → an between Zm and G. For all x, y ∈ Zn,

(x + y)Zn = ((x)Zn + (y)Zn )Zn f(x + y) = ax+y = axay = f(x)f(y).

n1−n2 For any n1, n2 ∈ Zm such that f(n1) = f(n2), a = e, m | n1 − n2 and |n1 − n2| ≤ m − 1. Then n1 = n2, f is injective. For any g ∈ G, there exists n ∈ N such that an = g. According to Division Theorem, there exist q, r ∈ N such that n = qm + r (0 ≤ r < m). Then g = an = aqm+r = (am)qar = eqar = ar = f(r) with 0 ≤ r < m for all g ∈ G, f is surjective. Hence there is a bijection between Zm and G, G is isomorphic to Zm. 8 DINGDING DONG

If G 6= {e} and is infinite, then there exists a 6= e ∈ G with an 6= e for all n ∈ N. Consider the linear map f : n → an between Z and G. For any x, y ∈ Zn, f(x + y) = ax+y = axay = f(x)f(y).

n1 n2 n1−n2 For any n1, n2 ∈ N with n1 6= n2, a 6= a (or a = e), so f is injective. For any g ∈ G, there exists n ∈ N such that an = g, so f is surjective. Hence there is a bijection f between Z and G, G is isomorphic to Z. Hence all cyclic groups are isomorphic either to Zn or to Z.  Theorem 5.4. A subgroup of a cyclic group is cyclic. Proof. Let G be a cyclic group generated by a and H ≤ G. If H = {e}, then H is a cyclic group generated by e. If H 6= {e}, then there exists a 6= e ∈ H and n ∈ N such that an ∈ H. Let m be the smallest positive integer such that am ∈ H. Since H is a subgroup of G, for any h ∈ H, there exists n ∈ N such that an = h. According to Division Theorem, there exist q, r ∈ N such that n = qm + r (0 ≤ r < m). Then an = aqm+r = (am)qar ar = (a−m)qan ar ∈ H by closure. Since 0 ≤ r < m and m is the smallest positive integer such that am ∈ H, r = 0. Hence n = qm, h = an = (am)q for all h ∈ H. m m Hence H is generated by a , H = ha i is cyclic. 

Theorem 5.5. For any n ∈ N, Zn has ϕ(n) generators, in which ϕ(n) = |{i | i ∈ N, 0 ≤ i < n, gcd(i, n) = 1}|.

Proof. For any i ∈ Zn, since Zn = {0, 1, ..., n − 1}, i ∈ N and 0 ≤ i < n. k k If i ∈ Zn and gcd(i, n) 6= 1, then for any k ∈ N, i ≡ ki(mod n), gcd(i, n)|i , k i 6= 1, i does not generate Zn. 2 n If i ∈ Zn and gcd(i, n) = 1, then it can be shown that i, i , ...i are all distinct. If there exist 1 ≤ k, l ≤ n such that ik = il, then (k − l)i ≡ 0(mod n). Since lcm(n, i) = ni,(k−l)i is a multiple of ni, k−l is a multiple of n. Since |k−l| ≤ n−1, 2 n |k − l| = 0, k = l. Since |Zn| = n, {i, i , ...i } = Zn, i generates Zn. Hence i is a generator of Zn if and only if gcd(i, n) = 1, Zn has ϕ(n) generators. 

6. Understanding Sets within Group Structures According to Theorem 2.11, a · a = a for a infinite, which explains the fact that R and C = {a + bi | a, b ∈ R} have the same cardinality. Moreover, the sets of real and complex numbers are also isomorphic as additive groups. Lemma 6.1. If there is a bijection between the bases of two vector spaces over the same field, then the two vector spaces are isomorphic as additive groups. ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY 9

Proof. Assume that there is a bijection between the bases of vector spaces V,W over field F. Let f : Bv → Bw be a bijective function between the bases of V and W . Then there exists h : V → W such that for any v = λ1v1 + ... + λnvn ∈ V , h(v) = λ1f(v1) + ... + λnf(vn) ∈ W . If there exist v = λ1v1+...+λnvn, u = µ1v1+...+µnvn ∈ V such that h(v) = h(u), then λ1f(v1) + ... + λnf(vn) = µ1f(v1) + ... + µnf(vn). Since each element in W can be uniquely written in a linear of , λi = µi for all i, v = u. Hence h is injective. Since for any w = λ1w1 + ... + λnwn ∈ W , there exists −1 −1 v = λ1f (w1) + ... + λnf (wn) ∈ V such that h(v) = w, h is also surjective and therefore bijective. Since for any v = λ1v1 + ... + λnvn, u = µ1v1 + ... + µnvn ∈ V , there is

h(v + u) = h(λ1v1 + ... + λnvn + µ1v1 + ... + µnvn)

= λ1f(v1) + ... + λnf(vn) + µ1f(v1) + ... + µnf(vn) = h(v) + h(u), h preserves the additive operations of V and W . Hence h : V → W is a bijection between V and W that preserves , V and W are isomorphic as additive groups.  Lemma 6.2. There is a bijection between the bases of R and C over Q.

Proof. Let B be a basis of R over Q. Since dimQ(R) is infinite, card(B) is infinite. Since B ⊆ R, iB ⊆ iR, B ∩iB = ∅. Since B ∪iB is a basis of C over Q, B ∩iB = ∅, card(B ∪ iB) = card(B)+ card(B) = card(B). Hence there is a bijection between the bases of R and C over Q.  Theorem 6.3. R and C are isomorphic as additive groups. Proof. According to Lemma 6.1 and Lemma 6.2, since there is a bijection between the bases of R and C over the same field Q, R and C are isomorphic as additive groups. 

Every set admits some group structure. With previous theorems about cardinal- ity, the bijection between X and F , the set of all finite subsets of X, helps construct a group structure within the set X. Lemma 6.4. For an infinite set X, there exists a bijection between X and F , in which F is the set of all finite subsets of X.

Proof. Since X is infinite, ℵ0 ≤ |X| by Theorem 2.7. For n = 0, 1, 2, ..., let Fn ⊆ F be the set of all finite subsets of |X| with cardinality S n n. Then F = n∈ Fn. Since there are |X| ways to choose n elements, each of N n n which from a group with cardinality |X|, |Fn| ≤ |X| . Since |X| = |X| according to Theorem 2.11, |Fn| ≤ |X|. Since F0,F1,F2, ... are disjoint from each other and S S F = Fn, |F | = |Fn| ≤ ℵ0|X| = |X| by Theorem 2.11. n∈N n∈N Meanwhile, since |X| = |{{x} | x ∈ F }| ≤ |F |, |X| = |F |. Hence X and F are of the same cardinality, there exists a bijection f : X → F between X and F .  10 DINGDING DONG

Lemma 6.5. F admits a group structure. Proof. For all f, g ∈ F , let f · g=(f \ g) ∪ (g \ f). Since for any f, g ∈ F , f ⊆ X, g ⊆ X,(f \g)∪(g\f) ⊆ (f ∪g) ⊆ X, f ·g is closed in F . For any f, g, h ∈ F , both (f · g) · h and f · (g · h) consist of elements which either are in exactly one of A, B or C, or are in all three, so f · g is associative. Let ∅ denote the , a perfectly good finite set. Since for any f ∈ F , f · ∅ = (f \ ∅) ∪ (∅ \ f) = f, there exists an identity element ∅ ∈ F . Since for any f ∈ F , there exists f ∈ F such that f · f = (f \ f) ∪ (f \ f) = ∅, so there exists an inverse element ∅ ∈ F . Hence hF, ·i is a group.  Theorem 6.6. Every set admits a group structure. Proof. Let X be an arbitrary set. If |X| is finite, then |X| = n for some n ∈ N, |X| = |Zn|. Then there exists a bijection f : X → Zn. Since hZn, +i is a cyclic group, hX, ◦i with −1 x1 ◦ x2 = f (f(x1) + f(x2)) is a cyclic group. If |X| is infinite, then according to Lemma 6.4 and Lemma 6.5, there exists a bijection f : X → F between X and F with hF, ·i being a group. Hence hX, ◦i is a group in which −1 x1 ◦ x2 = f (f(x1) · f(x2)). 

Acknowledgments. It is a pleasure to thank my mentor Brian McDonald for teaching me and offering suggestions for my paper. I would also like to thank Professor Peter May for organizing the REU program at the University of Chicago and Professor L´aszl´oBabai for teaching the apprentice program.

References [1] John B. Fraleigh. A First Course in . Pearson, 2003. [2] Gabriel Nagy. Cardinal Arithmetic. Available at author’s webpage. [3] Jonathan Pakianathan. Lecture Notes: and Cosets. Available at author’s webpage.