204Lecture22006 - Economics 204 Lecture 2 August 1 2006...

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Economics 204 Lecture 2, August 1, 2006 Revised 8/1/06, Revisions Marked by ** 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 . Let A m = f ( m ). We create an infinite matrix, whose ( m, n ) th entry is 1 if n A m , 0 otherwise: N 1 2 3 4 5 · · · A 1 = 0 0 0 0 0 · · · A 2 = { 1 } 1 0 0 0 0 · · · 2 N A 3 = { 1 , 2 , 3 } 1 1 1 0 0 · · · A 4 = N 1 1 1 1 1 · · · A 5 = 2 N 0 1 0 1 0 · · · . . . . . . . . . . . . . . . . . . . . . 1
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Now, on the main diagonal, change all the 0s to 1s and vice versa: N 1 2 3 4 5 · · · A 1 = 1 0 0 0 0 · · · A 2 = { 1 } 1 1 0 0 0 · · · 2 N A 3 = { 1 , 2 , 3 } 1 1 0 0 0 · · · A 4 = N 1 1 1 0 1 · · · A 5 = 2 N 0 1 0 1 1 · · · . . . . . . . . . . . . . . . . . . . . . The coding on the diagonal represents a subset of N which differs from each of the A m , contradiction. More formally, let t mn = 1 if n A m 0 if n A m Let A = { m : t mm = 0 } . ( Aside: this is the set described by changing all the codings on the diagonal. ) m A t mm = 0 m A m 1 A 1 A 1 so A = A 1 2 A 2 A 2 so A = A 2 . . . m A m A m so A = A m 2
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Therefore, A = 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 equiva- lent to R , so there are fundamentally more real numbers than rational numbers. Section 1.5: Algebraic Structures Field Axioms A field 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 num- bers; 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 defined. In fact, we define 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 num- bers; it need not be one in the real numbers. ) 8. Existence of multiplicative inverse: α F,α =0 !
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