Homework 3 Solution - MATH 74 HOMEWORK 2 (DUE WEDNESDAY...

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Unformatted text preview: MATH 74 HOMEWORK 2 (DUE WEDNESDAY SEPTEMBER 19) 1(a). * is commutative because for any x and y in (0 , ) we have x * y = xy x + y = yx y + x = y * x. (The outer two equalities hold by definition; the inner holds because multiplication and addition are commutative operations on real numbers.) 1(b). * is associative because for any x, y, z in (0 , ) we have x * ( y * z ) = x * yz y + z by definition = x yz y + z x + yz y + z by definition = xyz x ( y + z ) + yz after multiplying by y + z y + z = 1 = xyz xy + xz + yz and also ( x * y ) * z = xy x + y * z by definition = xy x + y z xy x + y + z by definition = xyz xy + z ( x + y ) after multiplying by x + y x + y = 1 = xyz xy + yz + xz Staring at these two a moment we see that indeed x * ( y * z ) = ( x * y ) * z . 1(c). There are no left identity elements for * . 1(d). There are no right identity elements for * . 2(a). * is commutative because for any x and y in Z we have x * y = 2( x + y ) = 2( y + x ) = y * x....
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Homework 3 Solution - MATH 74 HOMEWORK 2 (DUE WEDNESDAY...

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