By Hefferon J.

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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.

First, the two voters could be opposites, resulting after cancellation in the trivial election (with the majority cycle of all zeroes). Second, the two voters could have the same spin but come from different rows, as here. 1· D −1 voter T 1 voter R +1· D 1 voter T 1 voter −1 voter +0· R 1 voter T D 1 voter R 0 voters = T −1 voter 1 voter D 0 voters R 2 voters (b) There are two cases. , half the voters are D > R > T and half are T > R > D. Then cancellation gives the trivial election. If the number of voters is greater than one and odd (of the form 2k + 1 with k > 0) then using the cycle diagram from the proof, −a D T a R a b + D T −b R b c + D R T −c −a + b + c c = T D a−b+c R a+b−c we can take a = k and b = k and c = 1.

Thus this set is a subspace. 48 Linear Algebra, by Hefferon (d) The natural generalization is that the inverse image of a subspace of is a subspace. Suppose that X is a subspace of W . Note that 0W ∈ X so the set {v ∈ V h(v) ∈ X} is not empty. To show that this set is closed under combinations, let v1 , . . , vn be elements of V such that h(v1 ) = x1 , . . , h(vn ) = xn and note that h(c1 · v1 + · · · + cn · vn ) = c1 · h(v1 ) + · · · + cn · h(vn ) = c1 · x1 + · · · + cn · xn so a linear combination of elements of h−1 (X) is also in h−1 (X).