Principles of Mathematics (Math 2450) A Ë@ Õæ Aë áöß @. X. @ 2014-2015 Èð B@ É Ë@ Chapter 6 Cardinal Numbers and Cardinal Arithmetic In this chapter we introduce the concept of cardinal numbers. The properties of cardinal numbers and their arithmetic are studied. 6.1 The concept of cardinal numbers We will not define a cardinal number, but we will introduce it as a primitive concept related to the size of sets. Rules of cardinal numbers C-1 Each set A is associated with a cardinal number, denoted by card(a), and for each cardinal number n there is a set A such that card(a) = n. C-2 card(a) = 0 if and only if A = φ. C-3 If A is a nonempty finite set, that is A N k for some k N, then card(a) = k. C-4 For any two sets A and B, card(a) = card(b) if and only if A B. Example 1. Show that 1 is a cardinal number. 1
Example 2. Find card(a) if A = {a 1, a 2,, a 100 }. Example 3. Show that card(n) = card(z). Exercise 6.1 (1-7) 2
6.2 Ordering of the cardinal numbers - The Schröder-Bernstein Theorem Definition. (Finite and transfinite cardinal numbers) (1) A cardinal number of a finite set is called a finite cardinal number. (2) A cardinal number of an infinite set is called a transfinite cardinal number. Example 1. 0 and 1 are finite cardinal numbers where card(n) is a transfinite cardinal number. Definition. (Ordering of the cardinal numbers) Let A and B be sets. (1) We say that card(a) is less than or equal to card(b), and write card(a) card(b), if A is equipotent to a subset of B. (2) We say that card(a) is less than card(b), and write card(a) < card(b), if card(a) card(b) and card(a) card(b). Example 1. Show that card(n) < card(r). Example 2. Prove that 2014 < card(n). 3
Example 3. Let A and B be two sets. Prove that card(a) card(b) if and only if there exits an injection f : A B. Lemma 5.1. If B is a subset of A and if there exists an injection f : A B, then there is a bijection h : A B. Theorem 5.1. (Schröder-Bernstein Theorem ) If A and B are sets such that A is equipotent to a subset of B and B is equipotent to a subset of A, then A and B are equipotent. Proof. 4
Corollary 5.1. If A and B are sets such that card(a) card(b) and card(b) card(a), then card(a) = card(b). Proof. Exercise 6.2 (1-9) 5
6.3 Cardinal number of a power set-cantor s Theorem Theorem 5.2. (Cantor s Theorem) If X is a set, then card(x) < card(p(x)). Proof. Remark. The significance of Theorem 6.2 is that it furnishes a way to construct a far-reaching sequence of new transfinite cardinal numbers. For example, Exercise 6.3 (1-4) card(r) < card(p(r)) < card(p(p(r))) < 6
6.4 Addition of cardinal numbers Definition. (Addition of cardinal numbers) Let a and b be cardinal numbers. The cardinal sum of a and b, denoted by a + b, is the cardinal number card(a B), where A and B are disjoint sets such that card(a) = a and card(b) = b. Theorem 5.3. (Addition of cardinal numbers is well defined) Let a and b be cardinal numbers. Then (1) there exist disjoint sets A and B such that card(a) = a and card(b) = b, (2) if A 1, B 1, A 2, and B 2 are sets such that card(a 1 ) = card(a 2 ), card(b 1 ) = card(b 2 ), A 1 B 1 = φ, and A 2 B 2 = φ, then card(a 1 B 1 ) = card(a 2 B 2 ). Proof. Example. Find the cardinal sum 3 + 5 of the cardinal numbers 3 and 5. Theorem 5.4. (Addition of cardinal numbers is commutative and associative ) Let x, y, and z be arbitrary cardinal numbers. (1) Commutativity: x + y = y + x. (2) Associativity: x + (y + z) = (x + y) + z. 7
Proof. Notation. We denote card(n) = ℵ 0 (read Aleph null) and card(r) = c. Example 1. Find ℵ 0 + ℵ 0. Example 2. Find the cardinal sum ℵ 0 + c. 8
Example 3. Prove that if a is any transfinite cardinal number, then a + 1 = a. Exercise 6.4 (1-12) 9
6.5 Multiplication of Cardinal Numbers Definition. (Multiplication of cardinal numbers ) Let a and b be cardinal numbers. The cardinal product of a and b, denoted by ab, is the cardinal number Card(A B), where card(a) = a and card(b) = b. Remark. The above definition of cardinal product ab is independent of the choice of representatives A and B. For if A X and B Y, then by Theorem 5.7 (A B) (X Y ). Example 1. Let x be an arbitrary cardinal number. Show that the cardinal product x1 = x. Example 2. Prove that 0x = 0 for any cardinal number x. Theorem 5.5. (Properties of cardinal product) Let x, y, and z be arbitrary cardinal numbers. (1) Commutativity: xy = yx. (2) Associativity: x(yz) = (xy)z. (3) Distributivity: x(y + z) = xy + xz. Proof. 10
Example 1. Find ℵ 0 ℵ 0. Example 2. Prove that cc = c, where c = card(r). Example 3. Prove that xy = 0 implies x = 0 or y = 0, where x and y are cardinal numbers. Example 4. Let x, y, and z be cardinal numbers. Prove that x = y xz = yz. Exercise 6.5 (1-9) 11