Homework #2
EE 503: Summer 2026
Assigned: 01 June
Due: Monday, 08 June at 12:00
BrightSpace Assignment: Homework 2
Argument
Daily Derivation
Prove Bayes Theorem for evidence \(E\) and a partition of hypotheses \(H_1, \ldots, H_n\).
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
Use truth tables to prove whether these propositional assertions are valid or invalid:
\([(P \vee Q) \rightarrow R] \Leftrightarrow [(P \rightarrow R) \wedge (Q \rightarrow R)]\).
\([P \rightarrow (Q \rightarrow R)] \Rightarrow [(P \rightarrow Q) \rightarrow R]\).
\((P \Leftrightarrow Q) \Leftrightarrow [(P \wedge Q) \vee (\sim P \wedge \sim Q)]\).
\([(P \Rightarrow Q) \wedge (Q \Rightarrow R)] \Rightarrow (P \Rightarrow R)\).
Problem 2
Use mathematical induction to prove these theorems for all positive integers \(n \ge 1\):
\(\displaystyle \sum_{k=1}^n k \cdot k! = (n+1)! - 1\).
\(13^n - 4^n\) is divisible by 9.
\(\displaystyle \sum_{k=1}^n F_k^2 = F_n \cdot F_{n+1}\), for the Fibonacci sequence \(F_1 = F_2 = 1\) and \(F_{n+2} = F_{n+1} + F_n\).
Problem 3
Prove or disprove:
\(2^{A \cup B} = 2^A \cup 2^B\).
\(\cap_\alpha 2^{A_\alpha} = 2^{\cap_{\alpha} A_\alpha}\) for arbitrary set collection \(\{A_\alpha\}\).
\(\sigma(\sigma(\mathcal{A})) = \sigma(\mathcal{A})\) for any \(\mathcal{A} \subset 2^\Omega\).
Problem 4
Let \(\Omega = \{p, q, r, s, t, u\}\). Find the sigma-algebra \(\sigma(\emptyset)\) that the empty set \(\emptyset\) generates. Find the sigma-algebra \(\sigma(\{q, t\})\). Find the sigma-algebra \(\sigma(\{q, t, u\})\). Find \(\sigma(\sigma(\{q\}) \cup \sigma(\{u\}))\).
Problem 5
We do not know how the Ancient Green engineer Archimedes proved his famous result \(\frac{265}{153} < \sqrt{3} < \frac{1351}{780}\) described in “Measurement of a Circle”. Use a proof by contradiction to show that \(\sqrt{3}\) is an irrational number.
Problem 6
Prove by induction that the \(n\) eigenvectors \(e_1, \ldots, e_n\) of the \(n\)-by-\(n\) matrix \(A\) are linearly independent if the \(n\) corresponding eigenvalues \(\lambda_1, \ldots, \lambda_n\) are distinct. The column vector \(e_k\) is an eigenvector of \(A\) with scalar eigenvalue \(\lambda_k\) if and only if \(A e_k = \lambda_k e_k\) and \(e_k\) is not the null vector. The \(p\) column vectors \(v_1, \ldots, v_p\) are linearly independent if and only if the following implication holds for all complex scalars \(c_1, \ldots, c_p\): \(\sum_{k=1}^{p} c_k v_k = 0\) implies that all the scalars \(c_k\) are zero: \(c_1 = \cdots = c_p = 0\).
Book
Leon-Garcia
Chapter 2: 30, 73-76
Gubner
Chapter 1: 43, 45-46, 49, 53-54, 60