Homework #1
EE 503: Summer 2026
Assigned: 27 May
Due: Monday, 01 June at 12:00
BrightSpace Assignment: Homework 1
Daily Derivation
Prove the Addition Theorem: \(P(A \cup B) + P(A \cap B) = P(A) + P(B)\).
Write your solutions to these homework problems. Submit Handout and Book separately on BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.
Handout
Problem 1
Consider the integer function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) such that \(f(n) = n^2 + 6\) for any integer \(n\) in the set of integers \(\mathbb{Z}\). Define the range subsets \(A = \{6, 7, 15, 22\} \subset \mathbb{Z}\) and \(B = \{7, 22\} \subset \mathbb{Z}\). 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 \(f\): \(f^{-1}(S) = \{z \in \mathbb{Z} : f(z) \in S\}\).
Find \(f^{-1}(A)\), \(f^{-1}(B)\), \(f^{-1}(A \cup B)\), and \(f^{-1}(A \cap B)\).
Verify the commutations \(f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)\) and \(f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)\).
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.
\(A \subset B\) if and only if \(A - B = \emptyset\).
\(A \, \triangle \, B = A^c \, \triangle \, B^c\), for the symmetric difference \(A \, \triangle \, B = (A - B) \cup (B - A)\).
\(A - (B - C) = (A - B) \cup (A \cap C)\).
Problem 3
For all functions \(f\) and all sets \(A\) and \(B\) prove or disprove the following statements.
\(f(A \cap B) \subset f(A) \cap f(B)\).
\(f^{-1}(A \, \triangle \, B) = f^{-1}(A) \, \triangle \, f^{-1}(B)\).
\(f(A - B) = f(A) - f(B)\).
Problem 4
Let \(X = \{\alpha, \beta, \gamma\}\). 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 \(X = \{\phi, \chi, \psi, \omega\}\). 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.
Book
Leon-Garcia
Chapter 2: 117(d)
Gubner
Chapter 1: 15 (page 50)