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ma8_wk2_monday_notes_2010

ma8_wk2_monday_notes_2010 - MATH 8 SECTION 1 WEEK 2...

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MATH 8, SECTION 1, WEEK 2 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Monday, Oct. 4th’s lecture, where we discussed whether we could define a meaningful concept of “size” for infinite sets. 1. Random Question This is slightly different from the version I posted in class: there, I forgot that we should allow polynomials with an infinite number of positive-indexed terms! Oops. The corrected version is posted below: Question 1.1. Let E = ( X i = -∞ a i x i : a i R ) ; in other words, E consists of all of the “generalized” polynomials over R (polynomi- als where you can have both positive and negative coefficients, and possibly infinitely many of either.) Let F be the subset of E made of those polynomials with only finitely many negative-power terms: in other words, let F = { p ( x ) E : p ( x ) has only finitely many nonzero coefficients attached to negative powers of x. } Define addition and multiplication as normal for elements in F : i.e. X i = -∞ a i x i ! + X i = -∞ b i x i ! = X i = -∞ ( a i + b i ) x i , X i = -∞ a i x i ! · X i = -∞ b i x i ! = X i = -∞ | X infty j = -∞ a j b i - j x i . Define an order relation on F by saying p ( x ) = X i = -∞ a i x i > 0 iff a i > 0 , where i is the smallest integer such that a i is a nonzero coefficient of p ( x ) . Show that this is an ordered field that contains N , in which N is bounded. 1
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2 TA: PADRAIC BARTLETT 2. Sizes of Infinity: Introduction What does it mean for two sets to be the same size? In the finite case, this question is rather trivial; for example, we know that the two sets A = { 1 , 2 , 3 } , B = { A, B, emu } are the same size because they both have the same number of elements – in this case, 3. But what about infinite sets? For example, look at the sets N , Q , R , C ; are any of these sets the same size? Are any of them larger? By how much? In the infinite case, the tools we used for the finite – counting up all of the elements – don’t work. In response to this, we are motivated to try to find another way to count: in this case, one that involves functions .
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