/Solutions

1. A mean statistics problem

1. You are asked to find the typical temperature for a particular time of year centered around day 0. Suppose you have collected daily high temperatures x_-n...x_n for 2n+1 days in a row (with x₀, the day 0 temperature, in the middle), but you are only allowed to report one "typical" temperature y. You will be punished for deviation from the typical temperature according to the formula p = ∑_i (x_i-y)². Given an explicit formula for the value of y that minimizes p.

2. Now suppose you are allowed to include a predicted daily increment z, so that the guess for day i is now y+iz. Give an explicit formula for the values of y and z that between them minimize p = ∑_i (x_i - (y+iz))².

2. Equivalence relations and functions

Let f:A→B be some function.

1. Show that the relation ~_f, given by the rule x~_fy if and only if f(x) = f(y), is an equivalence relation.

2. Show the converse: if ~ is an equivalence relation on A, there exists a set B and a function f:A→B such that x~y if and only if f(x) = f(y).

3. Partial orders

Let A = ℕ^k be the set of all sequences of natural numbers of length k. Given two elements x and y of A, let x≤y if and only if x_i≤y_i for all i.

1. Prove (by showing reflexivity, antisymmetry, and trasitivity) that ≤ as defined above is a partial order on ℕ^k.

2. Give examples of choices of k for which ≤ is/is not a total order.

3. Prove that (ℕ^k, ≤) contains no infinite descending chain, i.e. there is no infinite sequence a₀, a₁, a₂, ..., where a_i+1 < a_i for all i.

4. Now consider the set ℕ^ℕ of all infinite sequences of natural numbers, where x ≤ y is defined to mean x_i ≤ y_i for all i∈ℕ. Prove or disprove: (ℕ^ℕ, ≤) contains no infinite descending chain.

4. Lattices

1. Prove or disprove: In any lattice, x∨(y∧z) = (x∨y)∧(x∨z).
2. Prove or disprove: In any lattice, x≥y ⇒ x∧(y∨z) = (x∧z)∨y.