204Lecture22008Web - Economics 204 Lecture 2, July 29, 2008...

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Economics 204 Lecture 2, July 29, 2008 Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2 N , the set of all subsets of N ,is not countable. Proof: Suppose 2 N is countable. Then there is a bijection f : N 2 N .L e t A m = f ( m ). We create an infnite matrix, whose ( m, n ) th entry is 1 iF n A m ,0otherw ise : N 1 2 345 ··· A 1 = 0 0 000 A 2 = { 1 } 1 0 2 N A 3 = { 1 , 2 , 3 } 11 1 00 A 4 = N 111 1 1 A 5 =2 N 0101 0 . . . . . . . . . . . . . . . . . . . . . 1
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Now, on the main diagonal, change all the 0s to 1s and vice versa: N 1 2 345 ··· A 1 = 1 0 000 A 2 = { 1 } 1 1 2 N A 3 = { 1 , 2 , 3 } 11 0 00 A 4 = N 111 0 1 A 5 =2 N 0101 1 . . . . . . . . . . . . . . . . . . . . . The coding on the diagonal represents a subset of N which diFers from each of the A m , contradiction. More formally, let t mn = 1i f n A m 0i f n 6∈ A m Let A = { m N : t mm =0 } .( Aside: this is the set described by changing all the codings on the diagonal. ) m A t mm m 6∈ A m 1 A 1 6∈ A 1 so A 6 = A 1 2 A 2 6∈ A 2 so A 6 = A 2 . . . m A m 6∈ A m so A 6 = A m 2
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Therefore, A 6 = f ( m ) for any m ,so f is not onto, contradiction. Message: There are fundamentally more subsets of N than elements of N . One can show that 2 N is numerically equivalent to R , so there are fundamentally more real numbers than rational numbers. Section 1.5: Algebraic Structures Field Axioms A Feld F =( F, + , · ) is a 3-tuple consisting of a set F and two binary operations + , · : F × F F such that 1. Associativity of +: α,β,γ F ( α + β )+ γ = α +( β + γ ) 2. Commutativity of +: α,β F α + β = β + α 3. Existence of additive identity: ! 0 F α F α +0=0+ α = α ( Aside: This says that 0 behaves like zero in the real numbers; it need not be zero in the real numbers. ) 4. Existence of additive inverse: α F ! ( α ) F α α )=( α α =0 ( Aside: We wrote α α ) rather than α α because substraction has not yet been deFned. In fact, we deFne α β to be α β ) . ) 3
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5. Associativity of · : α,β,γ F ( α · β ) · γ = α · ( β · γ ) 6. Commutativity of · : α,β F α · β = β · α 7. Existence of multiplicative identity: ! 1 F α F α · 1=1 · α = α ( Aside: This says that 1 behaves like one in the real numbers; it need not be one in the real numbers. ) 8. Existence of multiplicative inverse: α F,α 6 =0 ! α - 1 F α · α 1 = α 1 · α =1 ( Aside: We defne α β = αβ 1 . ) 9. Distributivity of multiplication over addition: α,β,γ F α · ( β + γ )= α · β + α · γ The point is that any property that Follows From the defnition oF feld (the “±ield Axioms”) must apply to
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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204Lecture22008Web - Economics 204 Lecture 2, July 29, 2008...

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