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1. Let l_1 and l_2 be lines through the origin in R^2 that intersect in an angle pi/n, and let r_i be the reflection about l_i. Prove that r_1 and r_2 generate a dihedral group Dn.
2. The symmetric group S_3 operates on two sets U and V of order 3. Decompose the product set U x V into orbits for the "diagonal action" g(u, v) = (gu, gv), when
A) the operations on U and V are transitive
B) the operation on U is transitive, the orbits for the operation on V are {v_1} and {v_2, v_3}
3. Let G be the group of symmetries of a cube, including the orientation-reversing symmetries. Describe the elements of G geometrically.
Systems of equations can be solved by graphing or by using substitution or elimination. What are the pros and cons of each method? What circumstances would cause you to use a different method?
Show that if n is odd then the set of all n-cycles consists of two conjugacy classes of equal size in An. Let G be a transitive permutation group on the finite set A with |A|>1. Show that there is some g in G such that g(a) is not equal to a for al..
Explain how to derive the distance formula (assuming that the distance formula is not yet known), first in 2 dimentional and then in 3 dimentional
The sequence of late times
Suppose you estimate the quotient 472 ÷ 58 using 500 ÷ 50 and using 420 ÷ 60. Which estimate should give a high estimate? Which estimate should give a low estimate? Explain the reasoning.
Let 'y' be the value of a house (in thousands of dollars), 'x' is the years after it is purchased. Some pairs of values are listed in the table below.
What are the cylindrical coordinates of the point whose rectangular coordinates are x = - 4 , y= 2, and z = 4?
Is a directed graph weakly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph?
Important information about Find the probabilities, Suppose a family plans 6 children, and the probability that a particular child is a girl is ½.
Important information about Chi square test for Goodness of fit & Association (1) Records of randomly selected births were obtained and categorized according to the day of the week that they occurred. Because babies are unfamiliar with our schedul..
A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 46.5ft^3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.
A committee consists of 8 married couples. In how many ways can a subcommitee of 5 people be chosen so that at most one married couple
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