12-240/Classnotes for Thursday October 18: Difference between revisions
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{{12-240/Navigation}} |
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===Riddle Along=== |
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The game of 15 is played as follows. Two players alternate choosing cards numbered between 1 and 9, with repetitions forbidden, so that the game ends at most after 9 moves (or four and a half rounds). The first player to have within her/his cards a set of precisely 3 cards that add up to 15 wins. |
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Does this game have a winning strategy? What is it? Who wins, the first to move or the second? Why am I asking this question at this particular time? |
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[[Image:12-240-DeckOfCards.png|center]] |
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See also [https://media.library.utoronto.ca/play.php?DJ6CPFxByy2J&id=8503 a video] and the [https://cmc.math.ca/home/videos/game-of-15-and-isomorphisms/ transcript] of that video. |
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{{12-240:Dror/Students Divider}} |
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== Theorems == |
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1. If G generates, |G| <math>\ge \!\,</math> n and G contains a basis, |G|=n then G is a basis |
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2. If L is linearly independent, |L| <math>\le \!\,</math> n and L can be extended to be a basis. |L|=n => L is a basis. |
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3.W <math>\subset \!\,</math> V a subspace then W is finite dimensioned and dim W <math>\le \!\,</math> dim V |
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If dim W = dim V, then V = W |
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If dim W < dim V, then any basis of W can be extended to be a basis of V |
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Proof of W is finite dimensioned: |
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Let L be a linearly independent subset of W which is of maximal size. |
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Fact about '''N''' |
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: Every subset A of '''N''', which is: |
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1. Non empty |
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2. Bounded : <math>\exist \!\,</math> N <math>\in \!\,</math> '''N''', <math>\forall \!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> N |
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has a maximal element: an element m <math>\in \!\,</math> A, <math>\forall\!\,</math> a <math>\in \!\,</math> A, a <math>\le \!\,</math> m ( m + 1 <math>\notin \!\,</math> A ) |
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== class note == |
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<gallery> |
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Image:12-240-Oct-15-Page-1.jpg |page1 |
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Image:12-240-Oct-15-Page-2.jpg |page2 |
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Image:12-240-Oct-15-Page-3.jpg |page3 |
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</gallery> |
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== lecture note on oct 18, uploaded by [[User:starash|starash]]== |
== lecture note on oct 18, uploaded by [[User:starash|starash]]== |
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Revision as of 19:20, 24 October 2012
lecture note on oct 18, uploaded by starash
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