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Intermediate Algebra
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Chapter AD / Exercise 11
Intermediate Algebra
Mckeague
Expert Verified
Discrete Structures Beifang Chen
We have textbook solutions for you!
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Intermediate Algebra
The document you are viewing contains questions related to this textbook.
Chapter AD / Exercise 11
Intermediate Algebra
Mckeague
Expert Verified
Contents Chapter 1. Set Theory 5 1.1. Sets and Subsets 5 1.2. Set Operations 6 1.3. Functions 9 1.4. Injection, Surjection, and Bijection 11 1.5. Infinite Sets 13 1.6. Permutations 15 Chapter 2. Number Theory 19 2.1. Divisibility 19 2.2. Greatest Common Divisor 20 2.3. Least Common Multiple 23 2.4. Modulo Integers 24 2.5. RSA Cryptography System 26 Chapter 3. Propositional Logic 29 3.1. Statements 29 3.2. Connectives 29 3.3. Tautology 31 3.4. Methods of Proof 33 3.5. Mathematical Induction 35 3.6. Boolean Functions 35 Chapter 4. Combinatorics 39 4.1. Counting Principle 39 4.2. Permutations 39 4.3. Combination 42 4.4. Combination with Repetition 45 4.5. Combinatorial Proof 46 4.6. Pigeonhole Principle 50 4.7. Relation to Probability 51 4.8. Inclusion-Exclusion Principle 53 4.9. More Examples 57 4.10. Generalized Inclusion-Exclusion Formula 59 Chapter 5. Recurrence Relations 63 5.1. Infinite Sequences 63 5.2. Homogeneous Recurrence Relations 64 5.3. Higher Order Homogeneous Recurrence Relations 67 5.4. Non-homogeneous Equations 68 3
4 CONTENTS 5.5. Divide-and-Conquer Method 69 5.6. Searching and Sorting 75 5.7. Growth of Functions 75 Chapter 6. Binary Relations 77 6.1. Binary Relations 77 6.2. Representation of Relations 79 6.3. Composition of Relations 80 6.4. Special Relations 82 6.5. Equivalence Relations and Partitions 84
CHAPTER 1 Set Theory 1.1. Sets and Subsets A set is a collection of objects satisfying certain properties; the objects in the collection are called elements (or objects or members ). A set is considered to be a whole entity and is different from its elements. Given a set A ; we write “ x A to say that x is an element of A or x belongs to A , and write “ x / A ” to say that x is not an element of A or x doesn’t belong to A . We usually denote sets by uppercase letters such as A, B, C, . . . , X, Y, Z , and denote the elements of a set by lowercase letters such as a, b, c, . . . , x, y, z , etc. There are two ways to express a set. One way is to list all elements of the set; the other way is to point out the attributes of the elements of the set. For example, let A be the set of integers whose absolute values are less than or equal to 3. The set A can be described in two ways: A = {- 3 , - 2 , - 1 , 0 , 1 , 2 , 3 } or A = { a | a is an integer, | a | ≤ 3 } . A set X whose elements satisfying Property P is denoted by X = { x | x satisfies P } or X = { x : x satisfies P } . In this note, most of time we use the first notation X = { x | x satisfies P } , and occasionally use the second notation when there is confusion to use the symbol “ | ”. There are two important things to be noticed about the concept of sets. The first one is that any set, when it is considered as an object, can not be an element of itself, but can be an element of another set. The second one is that for a particular object, it is possible to decide in principle whether or not the object is an element of a given set.

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