cisc102 Discrete Math Portfolio
Discrete Math Portfolio
Part 1: Topic Review
For each Unit taught this term, write a short summary of the topic. You should include relevant definitions, key information, and/or information on how to solve problems from this unit. A
summary of all topics covered is provided at the end of this document. It is recommended you review your weekly check-ins. Be sure to clearly highlight and discuss the topics within the unit you have mastered and those that you are still learning.
Requirements:
• Between 250-600 words for each section
• Keywords from each section are written in bold
• Most definitions from the unit are present
• The summary is written in your own words
• Must be written in LATEX(See Overleaf Template in onQ)
• Include helpful reminders that are particular to YOU (eg. “I always mix up equivalence relations and partial orders, so I want to be sure to remember those properties”).
Sample Response for Part 1:
Topic: Set Theory (Unit 1)
A set is a mathematical model for a collection of different things. The members of a set are called elements; two sets are equal if they contain the exact same elements. Sets may be finite or infinite. They are unordered and typically do not contain duplicate elements. We say U is the set of all elements within some domain, and ∅ is the set that contains nothing. Common operations include the union, intersection, difference, and complement. For sets A and B, each is written:
• Union: A ∪ B denotes the set of all elements in A or B. This includes elements that are in both A and B.
• Intersection: A ∩ B denotes the set of all elements in both A and B.
• Set Difference: A\B denotes the set of all elements in A that are not in B.
• Complement: A denotes the set of all elements that are not in A.
• Subsets: A ⊆ B if all the elements of A are also in B. If A = B then A is a proper subset of B (denoted A ⊂ B.
Some helpful reminders: You can remember that ∩ is the iNtersection because it looks like an n, and ∪ is the Union because it looks like a u. Applying the rules to simplify can be tricky, so go slow and state every rule you apply. Using a Venn Diagram can help understand the operations on sets.
Part 2: Problem Solving
For each unit, some difficult problems are provided below. Please write a full solution in LATEXto
FIVE problems from each section.
SETS and LOGIC
1. (a) Find the power set of S = {∅, a{a, b}}
(b) Find the power set of A × B where A = {a, b} and B = {{c}}
2. (a) Let A, B, C be sets such that |A| = x, |B| = y, |C| = z. Determine exactly or state
a range of values:
(i) |A ∪ B| (ii) |C\(A ∪ B)|
3. NEW Use a truth table to prove or disprove the following equivalence:
p → (q ∨ ¬r) ≡ ¬(p ∧ r) ∨ q
4. Let A = {n ∈ Z | n ≡ 3 (mod 4)} and let B = {n ∈ Z |
n−7 4 ∈ Z}. Prove that A = B.
5. Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you, “I had either popcorn or raisins. Also, if I had cucumber sandwiches, then I had soda. But I
didn’t drink soda or tea.” Of course you know that Tommy is the worlds worst liar, and everything he says is false. What did Tommy eat? Justify your answer by writing all of Tommy’s statements using sentence variables (P, Q, R, S, T), taking their negations, and using these to deduce what Tommy actually ate.
6. Let B(x) represent that person x is in Book Club. Let R(x, y) represent that person x has read book y. Express each logical phrase in plain language, and each plain language
statement in logical notation.
• No person in book club has read every book
• There is a book that no one in book club has read.
• ∀y ∃x such that (B(x) ∧ R(x, y))
• ∃x such that ∀y B(x) ∧ ¬R(x, y)
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