Chapter2 - Chapter 2 From Finite to Uncountable Sets A considerable amount of the material offered in this chapter is a review of terminology and

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Chapter 2 From Finite to Uncountable Sets A considerable amount of the material offered in this chapter is a review of termi- nology and results that were covered in MAT108. Our brief visit allows us to go beyond some of what we saw and to build a deeper understanding of some of the material for which a revisit would be bene ¿ cial. 2.1 Some Review of Functions We have just seen how the concept of function gives precise meaning for binary operations that form part of the needed structure for a ¿ eld. The other “big” use of function that was seen in MAT108 was with de ¿ ning “set size” or cardinality. For precise meaning of what constitutes set size, we need functions with two additional properties. De ¿ nition 2.1.1 Let A and B be nonempty sets and f : A ±² B. Then 1. f is one-to-one , written f : A 1 ± 1 B, if and only if ± 1 x ²± 1 y 1 z ²±± x ³ z ² + f F ± y ³ z ² + f " x ³ y ² ³ 2. f is onto , written f : A ± B, if and only if ± 1 y y + B " ± 2 x x + A F ± x ³ y ² + f ²²³ 3. f is a one-to-one correspondence , written f : A 1 ± 1 ± B, if and only if f is one-to-one and onto. 49
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50 CHAPTER 2. FROM FINITE TO UNCOUNTABLE SETS Remark 2.1.2 In terms of our other de ¿ nitions, f : A ±² B is onto if and only if rng ± f ² ³ def ´ y + B : ± 2 x ²± x + A F ± x ³ y ² + f ² µ ³ B which is equivalent to f [ A ] ³ B. In the next example, the ¿ rst part is shown for completeness and to remind the reader about how that part of the argument that something is a function can be proved. As a matter of general practice, as long as we are looking at basic functions that result in simple algebraic combinations of variables, you can assume that was is given in that form in a function on either its implied domain or on a domain that is speci ¿ ed. Example 2.1.3 For f ³ |t x ³ x 1 ± ¶ x u + U · U : ± 1 ´ x ´ 1 } , prove that f : ± ± 1 ³ 1 ² 1 ± 1 ± U . (a) By de ¿ nition, f l U · U ± i.e., f is a relation from ± ± 1 ³ 1 ² to U . Now suppose that x + ± ± 1 ³ 1 ² .Then x ´ 1 from which it follows that 1 ± ¶ x ¶ /³ 0 . Hence, ± 1 ± ¶ x ² ± 1 + U ± ´ 0 µ and y ³ x ¸ ± 1 ± ¶ x ² ± 1 + U because multiplication is a binary operation on U . Since x was arbitrary, we have shown that ± 1 x x + ± ± 1 ³ 1 ² " ± 2 y y + U F ± x ³ y ² + f ²² ± i.e., dom ± f ² ³ ± ± 1 ³ 1 ² . Suppose that ± x ³ y ² + f F ± x ³)² + f. Thenu ³ x ¸ ± 1 ± ¶ x ² ± 1 ³ ) because multiplication is single-valued on U · U .S i n c ex ³ u, and ) were arbitrary, ± 1 x 1 u 1 )²±± x ³ u ² + f F ± x + f " u ³ ± i.e., f is single-valued. Because f is a single-valued relation from ± ± 1 ³ 1 ² to U whose domain is ± ± 1 ³ 1 ² , we conclude that f : ± ± 1 ³ 1 ² ² U .
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2.1. SOME REVIEW OF FUNCTIONS 51 (b) Suppose that f ± x 1 ² ± f ± x 2 ² ± i.e., x 1 ³ x 2 + ± ² 1 ³ 1 ² and x 1 1 ² ³ x 1 ³ ± x 2 1 ² ³ x 2 ³ .
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This note was uploaded on 09/24/2008 for the course MAT 215 taught by Professor Pandhirapande during the Fall '08 term at Princeton.

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Chapter2 - Chapter 2 From Finite to Uncountable Sets A considerable amount of the material offered in this chapter is a review of terminology and

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