Homework #1

EE 364: Spring 2026

Assignment Details

Assigned: 13 January Due: Tuesday, 20 January at 16:00

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 0

Daily derivation #1, $P(A) + P(B) = P(A \cup B) + P(A \cap B)$.

Problem 1

Use truth tables to prove whether these propositional assertions are valid or invalid:

$(P \rightarrow \ \sim Q) \Leftrightarrow [(P \vee Q) \wedge \sim (P \vee Q)]$.
$\sim P \Rightarrow (P \Rightarrow Q)$.
$[P \wedge (Q \rightarrow R)] \Leftrightarrow [(\sim P \vee Q) \rightarrow (P \wedge R)]$.
$[(P \rightarrow Q) \wedge (P \rightarrow \ \sim Q)] \Leftrightarrow \ \sim P$.

Problem 2

Prove or disprove the following statements.

$[A \subset B] \Longrightarrow [A \cap B = A]$.
$[A^c \cup B = \Omega] \Longrightarrow [A \subset B]$.
$[A \subset B] \Longrightarrow [A^c \cup B = \Omega]$.

From (b) and (c) above can you justify that $[A \subset B]$ and $[A^c \cup B = \Omega]$ are equivalent? Two statements are equivalent if they always have the same truth value (either both true or both false but never different). Explain.

Problem 3

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.

$A \cap B = \emptyset$ if and only if $A \cup B = (A \cap B^c) \cup (A^c \cap B)$.
$(A \cap B) - C = (A - C) - (B - C)$.
$(A^c \cup B)^c \cap A^c = \emptyset$.

Problem 4

Suppose $\Omega = \{w, x, y, z\}$.

How many distinct subsets does $\Omega$ have? List them.
If $A = \{w, x\}$ how many supersets does $A$ have in $\Omega$? List them.
If $A = \{u, w, y\}$ how many supersets does $A$ have in $\Omega$? List them.

Problem 5

The sample space describes the set of all possible outcomes of a random experiment. The cardinality of $\left| \Omega \right|$ is the number of elements in $\Omega$. The power set $2^{\Omega} = \{A: A \subset \Omega\}$ is a collection of all the subsets of $\Omega$.

How big is the power set (i.e. cardinality, $\left| 2^\Omega \right|$) of each of the following sets?
1. $\Omega = \{H,T\}$.
2. The sample space represented by the outcome of a single roll of a six-sided die.
3. The set of positive integers less than or equal to $N$, $\{1, 2, \ldots, N\}$.
Write the power set for each of the following sample spaces:
1. $\Omega = \{a\}$.
2. $\Omega = \{a, b\}$.
3. $\Omega = \{a, b, c\}$.
Prove that if $A \subset B$ then $2^A \subset 2^B$.

Problem 6

Mary chooses a letter of the alphabet (a-z) at random. Describe the sample space $\Omega$ and probability measure $P$. Compute the probability that a vowel (a, e, i, o, u) is sampled.
A collection of plastic letters (a-z) is mixed in a jar. John draws two letters at random and without replacement (one after the other). Describe the sample space $\Omega$. What is the probability that John draws a vowel (a, e, i, o, u) and a consonant? What is the probability that John draws two vowels?

Problem 7

Events $A$ and $B$ have probabilities $P(A) = \frac34$ and $P(B) = \frac13$.

Show that $\frac{1}{12} \le P(A \cap B) \le \frac{1}{3}$.
Describe sample spaces and events $A$ and $B$ having $P(A) = \frac34$ and $P(B) = \frac13$ such that:
1. $P(A \cap B) = \frac{1}{12}$.
2. $P(A \cap B) = \frac{1}{3}$.

--- title: "Homework #1" subtitle: "EE 364: Spring 2026" format: html: toc: false number-sections: false code-fold: show code-tools: true theme: cosmo --- ::: {.callout-important} ## Assignment Details **Assigned:** 13 January **Due:** Tuesday, 20 January at 16:00 **BrightSpace Assignment:** [Homework 1](https://brightspace.usc.edu) ::: ::: {.callout-warning} ## 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 0 **Daily derivation #1**, $P(A) + P(B) = P(A \cup B) + P(A \cap B)$. ## Problem 1 Use truth tables to prove whether these propositional assertions are valid or invalid: a. $(P \rightarrow \ \sim Q) \Leftrightarrow [(P \vee Q) \wedge \sim (P \vee Q)]$. b. $\sim P \Rightarrow (P \Rightarrow Q)$. c. $[P \wedge (Q \rightarrow R)] \Leftrightarrow [(\sim P \vee Q) \rightarrow (P \wedge R)]$. d. $[(P \rightarrow Q) \wedge (P \rightarrow \ \sim Q)] \Leftrightarrow \ \sim P$. ## Problem 2 Prove or disprove the following statements. a. $[A \subset B] \Longrightarrow [A \cap B = A]$. b. $[A^c \cup B = \Omega] \Longrightarrow [A \subset B]$. c. $[A \subset B] \Longrightarrow [A^c \cup B = \Omega]$. From (b) and (c) above can you justify that $[A \subset B]$ and $[A^c \cup B = \Omega]$ are equivalent? Two statements are equivalent if they always have the same truth value (either both true or both false but never different). Explain. ## Problem 3 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. $A \cap B = \emptyset$ if and only if $A \cup B = (A \cap B^c) \cup (A^c \cap B)$. b. $(A \cap B) - C = (A - C) - (B - C)$. c. $(A^c \cup B)^c \cap A^c = \emptyset$. ## Problem 4 Suppose $\Omega = \{w, x, y, z\}$. a. How many distinct subsets does $\Omega$ have? List them. b. If $A = \{w, x\}$ how many supersets does $A$ have in $\Omega$? List them. c. If $A = \{u, w, y\}$ how many supersets does $A$ have in $\Omega$? List them. ## Problem 5 The sample space describes the set of all possible outcomes of a random experiment. The cardinality of $\left| \Omega \right|$ is the number of elements in $\Omega$. The power set $2^{\Omega} = \{A: A \subset \Omega\}$ is a collection of all the subsets of $\Omega$. a. How big is the power set (i.e. cardinality, $\left| 2^\Omega \right|$) of each of the following sets? i. $\Omega = \{H,T\}$. ii. The sample space represented by the outcome of a single roll of a six-sided die. iii. The set of positive integers less than or equal to $N$, $\{1, 2, \ldots, N\}$. b. Write the power set for each of the following sample spaces: i. $\Omega = \{a\}$. ii. $\Omega = \{a, b\}$. iii. $\Omega = \{a, b, c\}$. c. Prove that if $A \subset B$ then $2^A \subset 2^B$. ## Problem 6 a. Mary chooses a letter of the alphabet (a-z) at random. Describe the sample space $\Omega$ and probability measure $P$. Compute the probability that a vowel (a, e, i, o, u) is sampled. b. A collection of plastic letters (a-z) is mixed in a jar. John draws two letters at random and without replacement (one after the other). Describe the sample space $\Omega$. What is the probability that John draws a vowel (a, e, i, o, u) and a consonant? What is the probability that John draws two vowels? ## Problem 7 Events $A$ and $B$ have probabilities $P(A) = \frac34$ and $P(B) = \frac13$. a. Show that $\frac{1}{12} \le P(A \cap B) \le \frac{1}{3}$. b. Describe sample spaces and events $A$ and $B$ having $P(A) = \frac34$ and $P(B) = \frac13$ such that: i. $P(A \cap B) = \frac{1}{12}$. ii. $P(A \cap B) = \frac{1}{3}$.