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t 5 l EQUIVALENT SETS AND CARDINAL NUMBERS THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE By Shawing, Hsueh, B. S. Denton, Texas December, 1975 Hsueh, Shawing, Equivalent Sets and Cardinal Numbers. Master of Science (Mathematics), December, 19 7 5 , 38 pp., biblography, 4 titles. The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved. TABLE OF CONTENTS Chapter Page I. BASIC PROPERTIES ............... -.-.-......- 1 II. EQUIVALENT SETS............................. 10 III. CARDINAL NUMBERS............................29 BIBLIOGRAPHY..................-...-.-.-.....-.-.... 38 CHAPTER I BASIC PROPERTIES The purpose of this thesis is to study the equivalence relation between sets A and B: A ru B if and only if there exists a one to one function f from A onto B. The thesis is divided into three chapters: "Basic Properties," "Equivalent Sets," and "Cardinal Numbers." In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given and the Principle of Mathematical Induction and the Axiom of Choice are introduced. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied in detail and three different proofs of it are given. Some properties of sets of the power of the continuum are considered. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and a number of fundamental theorems on cardinal arithmetic are proved. 1.1. Definition. The Cartesian product of two sets A and B is the set of all ordered pairs (x,y) such that x c A and y F B. It is denoted by A x B. 1 2 1.2. Definition. Every subset of the Cartesian product A x B is called a relation from A to B. 1.3. Definition. The domain DR of a relation R is the set DR = {x : (x,y) 6 R}. 1.4. Definition. The range of a relation R is the set RR = {y : (x,y) R}. 1.5. Definition. A set f is called a function if and only if f is a relation and for every x,y,z, (x,y) 6 f and (x,z) C f imply y = z. 1.6. Definition. A function f is called one to one, if and only if for every x,x' 6 Df, f(x) = f(x') implies x = x' (or if for every x,x', (x,y) 6 f and (x',y) 6 f imply x = x'). 1.7. Definition. If f is a function, Df = A and Rf GB, then fCA x B and f is said to be a function from A into B. If Df = A and Rf = B, f is a function from A onto B. 1.8. Lemma. If f is a one to one function from A into B and g is a one to one function from B into C, then h = g(f) is a one to one function from A into C. Proof. It is obvious that h = g(f) is a function from A into C. Choose x,x' 6 A, and assume that g(f(x)) = h(x) = h(x') = g(f(x')). Since g is one to one, then f(x) = f(x'). Now, since f is one to one, it follows that x = x'. Thus h is a one to one function from A into C. 1.9. Lemma. If f is a function from A onto B and g is a function from B onto C, then h = g(f) is a function from A onto C. 3 Proof. Choose c e C. Since g is from B onto C, there exists some b E B such that g(b) = c. Further, since f is from A onto B, there exists some a F A such that b = f(a), and therefore, h(a) = g(f(a)) = g(b) = c. Thus h = g(f) is a function from A onto C. 1.10. Theorem. If f is a one to one function from A onto B and g is a one to one function from B onto C, then h = g(f) is a one to one function from A onto C. Proof. This result follows directly from Lemmas 1.8 and 1.9. 1.11. Definition. Let f be a function from A into B and let S be a subset of A. Define a new function g from S into B by g(x) = f(x) for x e S. This function g is called the restriction of f to S and is denoted by fI S. 1.12. Theorem. Let A,A2,B B2 be sets such that A1n A2 = q and B1F)B2 2=. (I)1ff isa one to one function from A onto B1 and f2 is a one to one function from A2 onto B2' then f = f1 U f2 (a common extension of f1 and f2) is a one to one function from A = A1U A2 onto B = B1U B2. (II) If f is a one to one function from A onto B and if f(A1 ) = B1 , then f, = fjA1 is a one to one function from A onto B1 and f 2 = fjA2 is a one to one function from A2 = A - A1 onto B 2 B - B1 . Proof. (I) Assume the hypotheses of (I). f(A) = f(A 1 U A2 ) Sf (A) U f (A2) = f1(A) U f2(A2) = B1 U B2 = B. Hence f is a function from A onto B. It remains to prove that f is one to one. Choose x1 ,x2 4 A and assume that f(x 1 ) = f(x2 ). Since f(A1 ) Af(A2 ) 1 2 = $, either both x1 and x2 are in A1 or both are in A2' Using the one to one property of f1 or of f2 x2 =x(2 ) follows. The proof of (II) is similar and is therefore omitted. 1.13. Definition. If A and B are sets, A is equivalent to B, denoted by A % B, means that there exists a one to one function from A onto B. I 1.14. Theorem. (I) A n A. 1 f f- (II) If A % B then B 'A. f g(f) (III) If A - B and B C then A C. Proof. (I) and (II) are obvious, and (III) follows by Theorem 1.10. 1.15. Notation. Let N = {1l,2,3,...,n,... 1, the set of natural numbers, and, for every n N, let N = {x : x 6 N and 1 x--n} = {1,2,3,...,n}. 1.16. Definition. A set S is finite means that either S = g or, for some n 6 N, S -v N n 1.17. Definition. A set S is denumerable means that S N. 1.18. Definition. A set S is countable means that either S is finite or S is denumerable. 1.19. Definition. A sequence of elements of a set B is a function from N into B and may be denoted by xn>. The range of the sequence <x > will denoted by'{x }. n n 1.20. Definition. A set S of natural numbers is called 5 inductive if n e S implies that n + 1 S. 1.21. The Mathematic alInduction Principle. If S is an inductive set and 1 e S, then S is the set of all natural numbers, that is, S = N. 1.22. Theorem. The following are equivalent: (I) A set S is countable; (II) A set S is either empty or the range of a sequence. Proof. (I) implies (II). If S is countable, then either (1) S = $, or (2) St N for some n 6 N, or (3) S N. If (1), n _1 the result is obvious. If (2), N S. Choose s e S and n1 define an extension of f~1 , say h, as follows: -.1 h(m) =f (m) if m c N ; h(m) = s if m 6 N but m i N. 1n Then h(N) = h(Nn) - -(Nn) = S, that is, h is a sequence whose range is S. If (3), f~1 is a one to one function from N onto S, and, therefore, f~1 is a sequence whose range is S. (II) implies (I). Assume S satisfies (II), that is, S is either empty or S is the range of a sequence. If S is empty, the result follows immediately. If S is the range of a sequence f, then S = f(N) = {X1 lx,x...,)Xn 2 ,. where xn = f(n). Clearly S is not empty. Case (1): S has finitely many distinct elements. In this case, it follows by Definition 1.16 that S ", N for some n E N.
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