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Discrete Mathematics and Its Applications


Synopsis


Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors. Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline.


McGraw-Hill Education's Connect, available as an optional add on item, is the only integrated learning system that empowers students by continuously adapting to deliver precisely what they need, when they need it and how they need it - ensuring class time is more effective. Connect allows the professor to assign homework, quizzes, and tests easily and automatically grades and records the scores of the student's work. Problems are randomized to prevent sharing of answers and may also have a "multi-step solution" which helps move the students' learning along if they experience difficulty.

Kenneth H. Rosen

Summary

Chapter 1: Logic

* Defines logical operators (AND, OR, NOT) and truth tables.
* Introduces propositional logic and logical equivalence.
* Explores predicate logic and quantifiers.

Real Example: Determining the validity of a logical argument about students passing a math exam:
* Premise: All students who study hard pass the math exam.
* Premise: John studies hard.
* Conclusion: John passes the math exam.
* Logical Argument: If (study hard) then (pass exam)
John studies hard
Therefore, John passes the exam

Chapter 2: Sets

* Defines sets, set operations (union, intersection, complement), and Venn diagrams.
* Covers set theory and cardinality.
* Explores applications in counting and probability.

Real Example: Creating a set of all odd numbers less than 10:
* Set: {1, 3, 5, 7, 9}
* Operation: Union with the set of even numbers less than 10 gives {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Chapter 3: Functions

* Defines functions, domain and range, and function notation.
* Covers types of functions (injective, surjective, bijective).
* Explores applications in computer science and mathematics.

Real Example: Representing the relationship between Celsius and Fahrenheit temperature:
* Function: f(C) = (9/5)*C + 32
* Domain: Set of all Celsius temperatures
* Range: Set of all Fahrenheit temperatures

Chapter 4: Counting and Probability

* Introduces counting principles (multiplication, addition).
* Covers permutations, combinations, and tree diagrams.
* Explores probability theory and conditional probability.

Real Example: Calculating the number of different three-digit numbers that can be formed using the digits 1, 2, 3, and 4:
* Permutations: 4 x 3 x 2 = 24
* Combinations: Not applicable for this scenario

Chapter 5: Mathematical Induction

* Defines mathematical induction and its steps.
* Covers applications in proving statements about natural numbers.
* Explores recurrence relations and generating functions.

Real Example: Proving that the sum of the first n odd numbers is a perfect square:
* Base Case: n = 1 (True: 1 is a perfect square)
* Inductive Hypothesis: Assume for some k > 1 that the sum of the first k odd numbers is a perfect square.
* Inductive Step: Show that the sum of the first (k + 1) odd numbers is also a perfect square.