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...xn 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 (xi-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 (xi - (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~fy 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 = ℕk 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 xi≤yi for all i.
Prove (by showing reflexivity, antisymmetry, and trasitivity) that ≤ as defined above is a partial order on ℕk.
- Give examples of choices of k for which ≤ is/is not a total order.
Prove that (ℕk, ≤) contains no infinite descending chain, i.e. there is no infinite sequence a0, a1, a2, ..., where ai+1 < ai for all i.
Now consider the set ℕℕ of all infinite sequences of natural numbers, where x ≤ y is defined to mean xi ≤ yi for all i∈ℕ. Prove or disprove: (ℕℕ, ≤) contains no infinite descending chain.
- 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.