posted Nov 2, 2009 11:13 AM by Adam Glesser
We took a field trip to the library to introduce everyone to the mathematics section. |
posted Oct 26, 2009 2:13 PM by Adam Glesser
We defined rings, commutative rings, division rings and fields. We then gave examples including the quaternion ring, rings of algebraic integrers and matrix rings. We also discussed Fermat's Last Theorem and the use and misuse of unique factorization. |
posted Oct 19, 2009 7:47 AM by Adam Glesser
We finished the discussion on groups by talking about Lagrange's theorem and its implications as well as the partial converse: Sylow's theorems. |
posted Oct 19, 2009 7:46 AM by Adam Glesser
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updated Oct 19, 2009 7:47 AM
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We explore the properties of the dihedral groups, proving that they are nonabelian groups of order 2n. We also defined subgroups of finite groups and discussed the classification of finite simple groups and Galois theory. |
posted Oct 4, 2009 7:34 PM by Adam Glesser
We prove that there exists a group of order n for any positive integer n, namely the cyclic group of order n. We started to define the dihedral groups. |
posted Sep 28, 2009 12:31 PM by Adam Glesser
We gave examples of quasigroups and loops, showing that there is a correspondence between quasigroups and latin squares. We then looked at some small examples of groups (order 1,2,3 and 4) and began discussing what is meant by an isomorphism. |
posted Sep 21, 2009 10:16 PM by Adam Glesser
We defined a magma as a set with a binary operation and computed the number of magmas that exist for a set with n elements. We then defined 5 properties of a binary operation we would like to have: associativity, divisibility, identity, inverse and commutativity. Using combinations of these properties, we defined semigroups, monoids, quasigroups, loops, groups and abelian groups. We gave examples of the first two and will give examples of the other 4 next time. |
posted Sep 21, 2009 10:16 PM by Adam Glesser
Today, we covered some of the basics of set theory, building off of our knowledge gained in Math 167 last spring. Topics included: naive sets versus Russell's paradox, notation (element of, subset, proper subset, etc.), the Cartesian product of sets, functions between sets, cardinality, Galileo's paradox. |
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