1873_solutions

1873_solutions

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Unformatted text preview: nd y unequal to 1 then is an associative commutative binary operation in the set R 1. It is clear that the operation is commutative. Now suppose that x, y and z are numbers unequal to 1. x y z  x yz x y z  xy xy  z  xyz xy xy xz xy z yz  xyz and we see similarly that x y z  x  y  z xy xz yz  xyz. An alternative way of looking at this exercise is to observe that if x and y are any numbers unequal to 1 then x y1 1 x 1 y and so x y z1 1 x y 1 z 1 1 2. Prove that if we define x1 y1 z. xy 1  xy is an associative commutative binary operation in y x for all numbers x and y in the interval 1, 1 then 1, 1 . We have already seen in an earlier exercise that is a binary operation in the set 1, 1 . It is clear that the operation is commutative. To see that is associative, suppose that x, y and z belong to 1, 1 . We observe that y z x  1yz x  y  z  xyz  x yz y z 1  yz  xy  xz 1  x 1yz and we can see similarly that x y z x  y  z  xyz 1  yz  xy  xz 3. Prove that if we define for all members x 1 , y 1 x1, y1 x2, y2  x1x2 y1y2, x1y2  x2y1 2 , then and x 2 , y 2 of R is an associative commutative binary operation in R 2 . 4. We can see the properties of this operation just as we did in the two preceding exercises. Alternatively, we can jump ahead to the material on complex numbers where this exercise appears again as part of the narrative. 5. (This exercise requires a little linear algebra.) Prove that if S is the set of all 2 2 matrices with real entries then matrix multiplication is an associative but not a commutative binary operation in S. The assertions made here are part of standard linear algebra. 53 6. Prove that if S is the set of all 2 2 matrices of the form a b ba where a and b are real numbers, then matrix multiplication is an associative commutative binary operation in S. We observe that if x 1 ,y 1 ,x 2 and y 2 then x1 y1 x2 y1 x1 y2  y2 x2 x1y2  y1x2 x1x2 y1y2 y1x2 x1y2 x1x2 and so, by relating ordered pairs x 1 , y 1 and matrices x1 y1 y1 x1 y1y2 we can see that this exercise is really Exercise 3 again in disguise. 7. Prove that if S is the set of all 2 2 matrices of the form cos  sin  sin  cos  where  is any real number, then matrix multiplication is an associative commutative binary operation in S. Once we have shown that the product of any two members of S must lie in S we shall know that this exercise is a special case of Exercises 3 and 5. We can think of the members of S as being the complex numbers whose distance from 0 is 1. Exercises on Finite Sets Many of the results you are asked to prove in the following exercises can be proved quite easily by mathematical induction. 1. Given that n is a positive integer and that S is a proper subset of the set 1, 2, , n , prove that card S  n. Solution: The assertion is obvious if n  1. We shall now assume that n  1 and we shall write n in the form m  1 where m is a positive integer. Suppose that S is a proper subset of the set 1, 2, , m  1 . In t...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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