Course Title: Discrete Mathematics for Engineers
Semester: Fall 2024
Chapter 1: Speaking Mathematically (1.1-1.3)
Chapter 2: Logic of Compound Statements (2.1-2.5)
Chapter 3: Logic of Quantified Statements (3.1-3.2)
Chapter 4: Elementary Number Theory and Methods of Proof (4.1-4.8)
Chapter 5: Sequences
Chapter 6: Set Theory (6.1-6.2)
6.1: Definitions - equality, Venn Diagrams, etc.
6.2: Properties of Sets
Chapter 7: Functions
Chapter 8: Relations
Chapter 9: Probability
Chapter 10: Graphs and Trees
9.3: Counting Elements of Disjoint Sets: The Addition Rule (Nov 11)
9.4: Pigeonhole Principle (Nov 11)
9.5: Counting Subsets of a Set: Combinations (Nov 11)
9.6: r-Combination with and without Repetition (brief) (Nov 11)
9.7: Pascal’s Formula and Binomial Theorem (Nov 11)
9.8: Probability Axioms and Expected Values (Nov 11)
9.9: Conditional Probability, Bayes Formula, and Independent Events (Nov 13)
Difference between Disjoint and Independent Events (Nov 13)
The number of subsets of size r from a set S is called r-combination.
Notable fact: Order in the subset is not important.
Set: {Ann, Bob, Cyd, Dan}
a. List all 3-combinations of the set:
{Bob, Cyd, Dan} (leave out Ann)
{Ann, Cyd, Dan} (leave out Bob)
{Ann, Bob, Dan} (leave out Cyd)
{Ann, Bob, Cyd} (leave out Dan)
b. Number of 3-combinations: 4
c. Ways to form a committee of size 3: 4
Ordered Selection: Important to consider the order of elements.
Unordered Selection: Only the identity of chosen elements matters.
Unordered selection of two elements from {0, 1, 2, 3}:
{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}
Total unordered selections: 6
Formula for choosing subsets of size r from n elements: (n choose r)
Choosing 5 members from 12: 792 distinct teams
Types of Poker Hands:
Royal Flush: 10, J, Q, K, A of the same suit
Straight Flush: Five adjacent denominations of the same suit
Other hands include Four of a Kind, Full House, etc.
Choose denominations for pairs
Choose cards from smaller denomination
Choose cards from larger denomination
Choose one from remaining cards
Total hands possible with two pairs: Computation based on combinatorics
Arrangements of letters in the word MISSISSIPPI
Distinguishable orderings are calculated based on steps outlined for placing letters.
Choosing r elements from n categories where repetition is allowed.
Formula: C(n+r-1, r)
Determine selections for 15 cans of soft drinks from 5 types
a. Total selections: C(19,15)
Consider cases where at least 6 cans of one type are chosen using the formulas above.
For the equation x1 + x2 + x3 + x4 = 10 with nonnegative integers, apply the selection formula.
Different situations for selection (order matters or not, repetition allowed or not) summarized in a table format for clear reference.