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92.322 Schedule You may use your notes, the handout, and
the photographs of the board during the final. You may not use books,
laptops, and PDAs. Office Hours for the week of May 18th: Monday, 6:15 - Tuesday, 6:15 - If enough people show up I will do a
review on Monday in the Mathematics department tutoring / study room. If not, I will answer concrete questions
that you might have. Here is
Midterm III (take-home version) Midterm III is on May 7 (not 5) Extra office
hours: Saturday, 2 May 2008, 4:30 -- No Office Hours
Today, May 2nd. |
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If a problem in ``Assigned and collected on this date"
column is underlined, click on it to see a solution.
All problems are due in one week unless stated otherwise.
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HOME | SYLLABUS | STUDENTS | SCHEDULE | MATERIALS | FORUM | LINKS |
Class attendance for the remainder of the course is required
Practice for the first midterm to
be held on March 5-th.
Solutions to the practice set (see
today's photos for the first two problems)
Practice for the second
midterm to be held on April 18
Practice set with answers for the third midterm to be held on May 5th
Class Number |
Topics /
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Assigned and collected on this date |
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1 |
Refresher: Sets, lists. Permutations. Factorials. |
given in class |
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2 |
Number of k-element subsets of an n-element set. Cube of dimension n. Board Photos. |
draw 4-dimensional cube |
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3 |
Binomial coefficients. Pascal's Triangle. Growth of binomial coefficients. Simplex of dimension n. Board Photos. |
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4 |
n-simplex and n-cube. Distributing identical gifts to distinguishable children. Fair distributions (each child gets at least one gift) – method of separators. |
What is the number of faces of n-cube (extra credit) ? |
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5 |
General distributions – method of borrowing and returning. General
formulation of combinatorial problems in terms of counting the number of ways
n balls can be put into k boxes. ``12-fold way" of balls can be identical or not – 2 cases; balls can be identical or not – 2 cases; each box may be required to hold at least one ball, at most one ball, or nor restrictions at all – 3 cases. Total is 12 cases. Interpreting the gift distribution problem in terms of boxes and balls. Counting the number of monomials of degree d that can be constructed from n variables. |
4-cube is collected |
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6 |
Counting the number of monomials of degree at most d that can be constructed from n variables. Formulation of Inclusion-Exclusion Principle. Notion of disjoint union. Notion of additive measure. |
assigned 1.8.26, 1.8.27, 1.8.33 |
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7 |
Example (2.5.5) of application of Inclusion-Exclusion Principle. Proof of I-E Principle based on the idea of canonical partition induced by the covering of the union by individual sets. Board Photos |
assigned 2.6.6 |
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8 |
Applications of Inclusion-Exclusion. The pigeonhole (box) principle. Board Photos |
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9 |
Euclidean Algorithm. Use of I-E principle to calculate Euler's function f(n). Board Photos |
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10 |
Practicing Euclidean Algorithm. Using Euclidean Algorithm to compute the multiplicative inverse of a number in modular arithmetic. Board Photos |
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11 |
Modular Arithmetic. Multiplicative Inverses. Board Photos |
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12 |
Multiplicative Inverses. Euclidean Algorithm. Group of Units. Generating Zm additively and Zm -0 multiplicatively. Board Photos |
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13 |
Groups of Units. Generating Zm additively and Zm -0
multiplicatively. |
prove that any number b that is NOT comprime with m does NOT generate Zm additively. |
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14 |
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15 |
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16 |
Board Photos (contains solutions to the first two problems of the practice set) |
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17 |
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18 |
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19 |
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20 |
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21 |
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14 March |
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31 March |
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2 April |
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4 April |
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7 April |
see the photos for the homework; due in a week. |
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9 April |
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11 April |
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14 April |
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16 April |
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23 April |
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28 April |
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30 April |
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1-5 May |
Board Photos: two lectures and the May 1st tutorial on graph representations given during Thursday office hours. |
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9 May |
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12 May |
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