Homework #1

EE 503: Fall 2025

Assignment Details

Assigned: 26 August
Due: Tuesday, 02 September at 12:00

BrightSpace Assignment: Homework 1

Instructions

Write your solutions to these homework problems. Submit your work to BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.

Problem 1

Consider the integer function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) such that \(f(n) = n^2 - 4\) for any integer \(n\) in the set of integers \(\mathbb{Z}\). Define the range subsets \(A = \{-4, -3, 5, 12\} \subset \mathbb{Z}\) and \(B = \{-3, 12\} \subset \mathbb{Z}\). Define the pullback or inverse image set \(f^{-1}(S)\) as the set of pre-images \(z \in \mathbb{Z}\) of \(S\) under the mapping \(f\): \(f^{-1}(S) = \{z \in \mathbb{Z} : f(z) \in S\}\).

  1. Find \(f^{-1}(A)\), \(f^{-1}(B)\), \(f^{-1}(A \cup B)\), and \(f^{-1}(A \cap B)\).

  2. Verify the commutations \(f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)\) and \(f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)\).

  3. Is it also true that \(f(A \cup B) = f(A) \cup f(B)\) and \(f(A \cap B) = f(A) \cap f(B)\)?

Problem 2

Prove or disprove the following statements. You must prove any set theoretic theorems that you use. That includes set associativity, commutativity, distributivity, De Morgan’s law, etc.

  1. \(A - (A - B) = A \cap B\).

  2. \(A \subset B\) if and only if \(A \cup B = B\).

Problem 3

For all functions \(f\) and all sets \(A\) and \(B\) prove or disprove the following statements.

  1. \(f^{-1}(A) \subset f^{-1}(B)\) if \(A \subset B\).

  2. \(f(A) \cap f(B) = f(A \cap B)\).

  3. \(f^{-1}(A - B) = f^{-1}(A) - f^{-1}(B)\).

Problem 4

Let \(X = \{a, b, c\}\). Find the power set \(2^X\). How many possible set collections \(\mathcal{A} \in 2^X\) are there? How many sigma-algebras can we define on \(X\)? Produce all of them. Produce a set collection that is not a sigma-algebra and then show how to minimally augment it to make it a sigma-algebra.

Problem 5

Let \(X = \{w, x, y, z\}\). Find the power set \(2^X\). How many possible set collections \(\mathcal{A} \in 2^X\) are there? Produce four set collections \(\mathcal{A} \subset 2^X\) that are not sigma-algebras. Show how to minimally augment these four set collections so that they are sigma-algebras.