# 1. A mean statistics problem

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 x_{0}, 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.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.

Show that the relation ~

_{f}, given by the rule x~_{f}y if and only if f(x) = f(y), is an equivalence relation.- 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.

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

^{k}.- Give examples of choices of k for which ≤ is/is not a total order.
Prove that (ℕ

^{k}, ≤) contains no infinite descending chain, i.e. there is no infinite sequence a_{0}, a_{1}, a_{2}, ..., where a_{i+1}< a_{i}for all i.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

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