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Let X be a set. a) X is infinite if and only if there is an injection X −→ P where P ⊆ X is a proper subset. b) X is infinite if and only if there is a surjection Q −→ X where Q ⊆ X is a proper subset. c) X is infinite if and only if there is an injection N0 −→ X. d) X is infinite if and only if there is a subset T ⊆ X and an injection N0 −→ T . 7. The set of all natural numbers N0 = {0, 1, 2, . } is infinite. Solution. Let us take the subset P = {1, 2, 3, . } and define a function f : N0 −→ P by f (n) = n + 1.

If we straighten out the paths starting at each number in the top row, so that we change the total number of crossings by 2 each time. So (−1)cσ +cτ = (−1)cτ σ . A permutation σ is called even if sgn σ = 1, otherwise it is odd. The set of all even permutations in Sn is denoted by An . Notice that ι ∈ An and in fact the following result is true. 8. The set An forms a group under composition. Proof. 7, if σ, τ ∈ An , then sgn(τ σ) = sgn(τ ) sgn(σ) = 1. Note also that ι ∈ An . The arrow diagram for σ −1 is obtained from that for σ by interchanging the rows and reversing all the arrows, so sgn σ −1 = sgn σ.

Let n be a positive integer. a) Prove the identities n+ n2 + 1 = 2n + ( n2 + 1 − n) = 2n + 1 √ . n + n2 + 1 √ √ b) Show that [ n2 + 1] = n and that the infinite continued fraction expansion of n2 + 1 is [n; 2n]. √ √ c) Show that [ n2 + 2] = n and that the infinite continued fraction expansion of n2 + 2 is [n; n, 2n]. PROBLEM SET 1 27 √ d) Show that [ n2 + 2n] = n and that the infinite continued fraction expansion of √ n2 + 2n is [n; 1, 2n]. 1-23. Find the fundamental solutions of Pell’s equation x2 − dy 2 = 1 for each of the values d = 5, 6, 8, 11, 12, 13, 31, 83.

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