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Answers for Topic: Voting Paradoxes 1 This is one example that yields a non-rational preference order for a single voter. 36 Linear Algebra, by Hefferon character most middle least Democrat Republican Third experience middle least most policies least most middle The Democrat is preferred to the Republican for character and experience. The Republican is preferred to the Third for character and policies. And, the Third is preferred to the Democrat for experience and policies.

19 Each of these projections is a homomorphism. Projection to the xz-plane and to the yz-plane are these maps.         x x x 0 y  →  0  y  → y  z z z z Projection to the x-axis, to the y-axis, and to the z-axis are these maps.             x x x 0 x 0 y  →  0   y  → y   y  → 0 z 0 z 0 z z And projection to the origin is this map.     x 0 y  → 0 z 0 Verification that each is a homomorphism is straightforward.

4 These checks can be done by listing all of the possibilities. For instance, to verify the commutativity of addition, that a + b = b + a, we can easily check it for all possible pairs a, b, because there are only four such pairs. Similarly, for associativity, there are only eight triples a, b, c, and so the check is not too long. ) 5 These will do. · 0 1 2 + 0 1 2 0 0 1 2 0 0 0 0 1 1 2 0 1 0 1 2 2 2 0 1 2 0 2 1 As in the prior item, the check that they satisfy the conditions can be done by listing all of the cases, although this way of checking is somewhat long (making use of commutativity is helpful in shortening the work).

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