CIS 1600: A Comprehensive Survival Guide

Navigating the challenges of CIS 1600: Mathematical Foundations of Computer Science can be daunting. However, conduct.edu.vn offers this survival guide, providing essential mathematical knowledge, proof techniques, and problem-solving methods. This resource equips you for success, enhancing your analytical abilities and preparing you for advanced computer science topics with insights, techniques, and strategies for mastering mathematical concepts crucial for computer science. This guide aims to help students with discrete mathematics, logical reasoning, and computational theory.

1. Understanding the Core Concepts of CIS 1600

CIS 1600 lays the groundwork for many advanced computer science topics. A firm grasp of its core concepts is crucial for success. This section will break down the fundamental ideas you’ll encounter, providing clear explanations and examples to solidify your understanding.

1.1. Propositional Logic: The Foundation of Reasoning

Propositional logic is the study of statements that can be either true or false. These statements, called propositions, can be combined using logical connectives to form more complex statements. Understanding propositional logic is essential for constructing and evaluating arguments, designing digital circuits, and verifying software.

Key Concepts:

• Propositions: Simple statements that are either true (T) or false (F). Examples: “The sky is blue,” “2 + 2 = 5.”
• Logical Connectives: Operators that combine propositions. Common connectives include:
  • Negation (¬): Reverses the truth value of a proposition. If P is true, then ¬P is false, and vice versa.
  • Conjunction (∧): “And.” P ∧ Q is true only if both P and Q are true.
  • Disjunction (∨): “Or.” P ∨ Q is true if either P or Q (or both) are true.
  • Implication (→): “If…then…” P → Q is false only if P is true and Q is false.
  • Biconditional (↔): “If and only if.” P ↔ Q is true if P and Q have the same truth value (both true or both false).
• Truth Tables: Tables that show the truth value of a compound proposition for all possible combinations of truth values of its constituent propositions.
• Tautologies, Contradictions, and Contingencies:
  • Tautology: A proposition that is always true, regardless of the truth values of its variables.
  • Contradiction: A proposition that is always false.
  • Contingency: A proposition that is neither a tautology nor a contradiction; its truth value depends on the truth values of its variables.
• Logical Equivalence: Two propositions are logically equivalent if they have the same truth table.

Examples:

Let P be “It is raining” and Q be “The ground is wet.”

• ¬P: “It is not raining.”
• P ∧ Q: “It is raining and the ground is wet.”
• P ∨ Q: “It is raining or the ground is wet.”
• P → Q: “If it is raining, then the ground is wet.”
• P ↔ Q: “It is raining if and only if the ground is wet.”

Truth Table Example (Implication):

P	Q	P → Q
True	True	True
True	False	False
False	True	True
False	False	True

Why it matters: Propositional logic provides a framework for formalizing reasoning. It’s used extensively in computer science for tasks such as:

• Validating arguments: Determining whether an argument is logically sound.
• Designing digital circuits: Representing the behavior of logic gates.
• Verifying software: Ensuring that a program behaves as expected under all possible conditions.

1.2. Predicate Logic: Reasoning About Objects and Their Properties

Predicate logic extends propositional logic by allowing us to reason about objects and their properties. It introduces predicates, which are statements that can be true or false depending on the object they are applied to. This allows us to express more complex relationships and make more general statements than propositional logic.

Key Concepts:

• Predicates: Statements that assert a property about an object. They can take one or more arguments.
  • Example: isPrime(x) could be a predicate that is true if x is a prime number and false otherwise.
• Quantifiers: Symbols that specify the quantity of objects that satisfy a predicate.
  • Universal Quantifier (∀): “For all.” ∀x P(x) means that the predicate P(x) is true for all objects x in the domain.
  • Existential Quantifier (∃): “There exists.” ∃x P(x) means that there exists at least one object x in the domain for which the predicate P(x) is true.
• Variables: Symbols that represent objects in the domain of discourse.
• Domain of Discourse: The set of all objects that the variables can refer to.

Examples:

Let the domain of discourse be the set of all integers.

• Predicate: even(x) – “x is an even number.”
• ∀x even(x) → even(x+2): “For all integers x, if x is even, then x+2 is even.”
• ∃x prime(x) ∧ x > 10: “There exists an integer x such that x is prime and x is greater than 10.”
• ¬∃x odd(x) ∧ divisibleBy(x, 2): “It is not the case that there exists an integer x such that x is odd and divisible by 2.”

Why it matters: Predicate logic is fundamental to:

• Formalizing mathematical statements: Expressing theorems and definitions precisely.
• Verifying software: Specifying the desired behavior of programs and proving that they meet those specifications.
• Artificial intelligence: Representing knowledge and reasoning about the world.
• Database systems: Querying and manipulating data based on complex criteria.

1.3. Proof Techniques: Establishing Truth with Certainty

In mathematics and computer science, a proof is a rigorous argument that demonstrates the truth of a statement. Mastering various proof techniques is essential for verifying the correctness of algorithms, establishing mathematical theorems, and developing sound logical arguments.

Key Concepts:

• Direct Proof: Starting with the assumptions (premises) and using logical deductions to arrive at the conclusion.
• Proof by Contraposition: Proving P → Q by proving its contrapositive, ¬Q → ¬P. This is based on the logical equivalence of the two statements.
• Proof by Contradiction: Assuming the negation of the statement you want to prove and then deriving a contradiction. This shows that the initial assumption must be false, and therefore the original statement must be true.
• Proof by Induction: Used to prove statements about natural numbers. It involves two steps:
  • Base Case: Proving the statement for the smallest value (usually 0 or 1).
  • Inductive Step: Assuming the statement is true for some value k (the inductive hypothesis) and then proving that it must also be true for k+1.
• Proof by Cases: Dividing the possible scenarios into a finite number of cases and proving the statement for each case.

Examples:

• Direct Proof: Prove that if n is an even integer, then n² is an even integer.
  • Assume n is even. Then n = 2k for some integer k.
  • Therefore, n² = (2k)² = 4k² = 2(2k²).
  • Since 2k² is an integer, n² is even.
• Proof by Contraposition: Prove that if n² is an odd integer, then n is an odd integer.
  • We prove the contrapositive: If n is even, then n² is even (which we already proved above).
• Proof by Contradiction: Prove that √2 is irrational.
  • Assume √2 is rational. Then √2 = a/b, where a and b are integers with no common factors.
  • Squaring both sides, we get 2 = a²/b², so a² = 2b².
  • This means a² is even, so a is even (as shown above). Therefore, a = 2k for some integer k.
  • Substituting, we get (2k)² = 2b², so 4k² = 2b², and b² = 2k².
  • This means b² is even, so b is even.
  • But this contradicts our assumption that a and b have no common factors. Therefore, √2 must be irrational.
• Proof by Induction: Prove that the sum of the first n positive integers is n(n+1)/2.
  • Base Case (n=1): 1 = 1(1+1)/2 = 1.
  • Inductive Step: Assume the statement is true for n=k: 1 + 2 + … + k = k(k+1)/2.
  • We need to show that the statement is true for n=k+1: 1 + 2 + … + k + (k+1) = (k+1)(k+2)/2.
  • Starting with the left side: 1 + 2 + … + k + (k+1) = k(k+1)/2 + (k+1) (by the inductive hypothesis).
  • Simplifying: k(k+1)/2 + (k+1) = (k(k+1) + 2(k+1))/2 = (k+1)(k+2)/2.
  • This is the right side, so the statement is true for n=k+1.

Why it matters: Proof techniques are essential for:

• Verifying algorithms: Ensuring that an algorithm produces the correct output for all possible inputs.
• Establishing mathematical theorems: Proving the validity of mathematical statements.
• Developing sound logical arguments: Constructing persuasive and reliable arguments in any field.

1.4. Set Theory: Organizing and Manipulating Collections of Objects

Set theory is the foundation of mathematics and computer science. It provides a framework for defining, organizing, and manipulating collections of objects. Understanding set theory is crucial for database design, algorithm analysis, and many other areas.

Key Concepts:

• Sets: Unordered collections of distinct objects.
  • Example: A = {1, 2, 3}
• Elements: The objects that belong to a set.
  • Example: 1 ∈ A (1 is an element of set A)
• Subset: A set whose elements are all contained within another set.
  • Example: B = {1, 2}. B ⊆ A (B is a subset of A)
• Proper Subset: A subset that is not equal to the original set.
  • Example: B = {1, 2}. B ⊂ A (B is a proper subset of A)
• Universal Set: The set containing all possible elements under consideration.
• Empty Set (∅): The set containing no elements.
• Set Operations:
  • Union (∪): The set containing all elements from both sets. A ∪ B = {x | x ∈ A or x ∈ B}
  • Intersection (∩): The set containing only the elements that are in both sets. A ∩ B = {x | x ∈ A and x ∈ B}
  • Difference (-) The set containing elements that are in the first set but not in the second set. A – B = {x | x ∈ A and x ∉ B}
  • Complement (A’): The set containing all elements in the universal set that are not in A. A’ = {x | x ∈ U and x ∉ A} (where U is the universal set)
• Power Set: The set of all possible subsets of a set.
  • Example: If A = {1, 2}, then the power set of A is P(A) = {∅, {1}, {2}, {1, 2}}

Examples:

Let A = {1, 2, 3} and B = {3, 4, 5}.

• A ∪ B: {1, 2, 3, 4, 5}
• A ∩ B: {3}
• A – B: {1, 2}
• If the universal set U = {1, 2, 3, 4, 5, 6}, then A’: {4, 5, 6}

Why it matters: Set theory is crucial for:

• Database design: Representing and manipulating data in relational databases.
• Algorithm analysis: Describing the input and output of algorithms.
• Formal language theory: Defining the syntax and semantics of programming languages.
• Probability theory: Defining events and sample spaces.

1.5. Functions and Relations: Mapping and Connecting Objects

Functions and relations are fundamental concepts in mathematics and computer science that describe how objects are related to each other. Functions provide a mapping from one set of objects to another, while relations describe general connections between objects.

Key Concepts:

• Relation: A set of ordered pairs. A relation from set A to set B is a subset of A × B (the Cartesian product of A and B).
  • Example: Let A = {1, 2} and B = {a, b}. A relation from A to B could be R = {(1, a), (2, b)}.
• Function: A special type of relation where each element in the domain (input set) is associated with exactly one element in the codomain (output set).
  • A function f: A → B maps each element x in set A to a unique element f(x) in set B.
• Domain: The set of all possible inputs to a function.
• Codomain: The set containing all possible outputs of a function.
• Range: The set of all actual outputs of a function (a subset of the codomain).
• Types of Functions:
  • Injective (One-to-one): A function where each element in the codomain is mapped to by at most one element in the domain. If f(x₁) = f(x₂) then x₁ = x₂.
  • Surjective (Onto): A function where every element in the codomain is mapped to by at least one element in the domain. For every y in B, there exists an x in A such that f(x) = y.
  • Bijective (One-to-one Correspondence): A function that is both injective and surjective.
• Composition of Functions: Applying one function after another. If f: A → B and g: B → C, then the composition g ∘ f: A → C is defined as (g ∘ f)(x) = g(f(x)).

Examples:

• Relation: Let A = {1, 2, 3} and B = {a, b, c}. The relation R = {(1, a), (2, b), (3, a)} is a relation from A to B.
• Function: Let A = {1, 2, 3} and B = {a, b, c}. The function f = {(1, a), (2, b), (3, c)} is a function from A to B. The domain is A, the codomain is B, and the range is {a, b, c}.
• Injective Function: Let f(x) = 2x from the set of integers to the set of integers. This is injective because each output is produced by at most one input.
• Surjective Function: Let f(x) = x mod 2 from the set of integers to the set {0, 1}. This is surjective because every element in the codomain {0, 1} is mapped to by at least one element in the domain (e.g., f(0) = 0, f(1) = 1).
• Bijective Function: Let f(x) = x + 1 from the set of integers to the set of integers. This is bijective because it is both injective and surjective.
• Composition of Functions: Let f(x) = x² and g(x) = x + 1. Then (g ∘ f)(x) = g(f(x)) = g(x²) = x² + 1.

Why it matters: Functions and relations are crucial for:

• Programming: Defining functions and data structures in programming languages.
• Database systems: Representing relationships between tables in relational databases.
• Algorithm design: Describing the behavior of algorithms and their inputs and outputs.
• Mathematical modeling: Representing real-world phenomena with mathematical equations.

1.6. Induction and Recursion: Building Structures and Solving Problems Iteratively

Induction and recursion are powerful techniques for defining structures and solving problems in computer science and mathematics. They allow us to build complex objects and algorithms from simpler ones by iteratively applying a set of rules.

Key Concepts:

• Recursion: A technique where a function calls itself within its own definition. It typically involves a base case (a simple case that can be solved directly) and a recursive step (where the problem is broken down into smaller subproblems).
  • Example: The factorial function can be defined recursively as:
    • factorial(0) = 1 (base case)
    • factorial(n) = n * factorial(n-1) (recursive step)
• Induction: A method of proving statements about natural numbers. It involves a base case (proving the statement for the smallest value) and an inductive step (assuming the statement is true for some value k and then proving that it must also be true for k+1).
  • Example: Proving that the sum of the first n positive integers is n(n+1)/2 (as shown in the proof techniques section).
• Inductive Definition: Defining a set or structure using a base case and an inductive step.
  • Example: The set of natural numbers can be defined inductively as:
    • 0 is a natural number.
    • If n is a natural number, then n+1 is a natural number.

Examples:

• Recursive Function (Factorial):

  def factorial(n):
      if n == 0:
          return 1
      else:
          return n * factorial(n-1)

• Inductive Definition (Binary Tree):

  • A single node is a binary tree.
  • If T₁ and T₂ are binary trees, then a node with T₁ as its left subtree and T₂ as its right subtree is also a binary tree.

Why it matters: Induction and recursion are essential for:

• Algorithm design: Developing efficient algorithms for searching, sorting, and other common tasks.
• Data structures: Defining and manipulating complex data structures like trees and graphs.
• Formal language theory: Defining the syntax and semantics of programming languages.
• Artificial intelligence: Developing reasoning systems and machine learning algorithms.

1.7. Combinatorics: Counting and Arranging Objects

Combinatorics is the branch of mathematics that deals with counting and arranging objects. It provides tools for determining the number of ways to choose, arrange, or combine objects according to specific rules. This is crucial for analyzing algorithms, designing experiments, and solving probability problems.

Key Concepts:

• Permutations: Arrangements of objects in a specific order. The number of permutations of n distinct objects is n! = n × (n-1) × (n-2) × ... × 2 × 1.

  • Example: The number of ways to arrange the letters “ABC” is 3! = 6 (ABC, ACB, BAC, BCA, CAB, CBA).

• Combinations: Selections of objects without regard to order. The number of combinations of choosing k objects from a set of n objects is given by the binomial coefficient:

  n! / (k! * (n-k)!)

  • Example: The number of ways to choose 2 letters from the set {A, B, C} is 3! / (2! * 1!) = 3 (AB, AC, BC).

• Basic Counting Principles:

  • Rule of Sum: If there are n ways to do one task and m ways to do another task, and the tasks cannot be done simultaneously, then there are n + m ways to do either task.
  • Rule of Product: If there are n ways to do one task and m ways to do another task, then there are n × m ways to do both tasks.

• Pigeonhole Principle: If n items are put into m containers, with n > m, then at least one container must contain more than one item.

Examples:

• Permutations: How many ways can you arrange 5 books on a shelf? 5! = 120 ways.
• Combinations: How many ways can you choose a committee of 3 people from a group of 10? 10! / (3! * 7!) = 120 ways.
• Rule of Sum: You can travel from city A to city B by car (3 routes) or by train (2 routes). How many ways can you travel from A to B? 3 + 2 = 5 ways.
• Rule of Product: You have 4 choices for a main course and 3 choices for a dessert. How many different meals can you create? 4 * 3 = 12 meals.
• Pigeonhole Principle: If you have 13 people, at least two of them must have been born in the same month.

Why it matters: Combinatorics is crucial for:

• Algorithm analysis: Counting the number of operations an algorithm performs.
• Probability theory: Calculating the probability of events.
• Cryptography: Designing secure codes and ciphers.
• Computer networks: Analyzing network traffic and routing algorithms.

1.8. Graph Theory: Modeling Relationships and Networks

Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. It provides a powerful tool for analyzing networks, relationships, and structures in various fields, including computer science, social science, and biology.

Key Concepts:

• Graph: A set of vertices (nodes) connected by edges.
  • Vertices (Nodes): Represent objects.
  • Edges: Represent relationships between objects.
• Types of Graphs:
  • Directed Graph: Edges have a direction (from one vertex to another).
  • Undirected Graph: Edges have no direction.
  • Weighted Graph: Edges have a weight associated with them (e.g., distance, cost).
  • Complete Graph: Every pair of vertices is connected by an edge.
  • Connected Graph: There is a path between every pair of vertices.
  • Acyclic Graph: A graph with no cycles.
• Path: A sequence of vertices connected by edges.
• Cycle: A path that starts and ends at the same vertex.
• Degree of a Vertex: The number of edges incident to a vertex.
• Graph Representation:
  • Adjacency Matrix: A matrix where the entry at row i and column j is 1 if there is an edge from vertex i to vertex j, and 0 otherwise.
  • Adjacency List: A list of neighbors for each vertex.
• Graph Traversal Algorithms:
  • Breadth-First Search (BFS): Visits all the neighbors of a vertex before moving to the next level of neighbors.
  • Depth-First Search (DFS): Explores as far as possible along each branch before backtracking.

Examples:

• Social Network: Vertices represent people, and edges represent friendships.
• Computer Network: Vertices represent computers, and edges represent network connections.
• Road Network: Vertices represent cities, and edges represent roads.
• Directed Graph: A website link structure, where vertices represent web pages and edges represent hyperlinks.
• Weighted Graph: A map where vertices are cities and edges are roads, with weights representing the distance between cities.

Why it matters: Graph theory is crucial for:

• Computer networks: Designing routing algorithms and analyzing network traffic.
• Social networks: Analyzing social connections and influence.
• Algorithm design: Developing algorithms for searching, sorting, and optimization.
• Database systems: Representing relationships between data entities.
• Artificial intelligence: Developing knowledge representation systems and planning algorithms.
• Operations Research: Finding optimal solutions to network flow problems, scheduling, and resource allocation.

2. Mastering Proof Techniques

As mentioned earlier, proof techniques are essential for CIS 1600. This section provides a deeper dive into various proof methods, offering strategies and examples to help you construct rigorous and convincing arguments.

2.1. Direct Proofs: Building a Logical Chain

Direct proofs are the most straightforward way to prove a statement. You start with the given assumptions (premises) and use logical deductions to arrive at the desired conclusion.

Strategy:

1. Understand the statement: Make sure you clearly understand what you need to prove.
2. Identify the assumptions: Determine what information is given to you.
3. Construct a logical chain: Use definitions, axioms, and previously proven theorems to connect the assumptions to the conclusion. Each step in the chain must be logically valid.
4. Write clearly and concisely: Present your proof in a clear and organized manner, explaining each step and justifying your reasoning.

Example:

Prove: If n is an odd integer, then n² is an odd integer.

Proof:

1. Assume n is odd.
2. By definition, an odd integer can be written in the form n = 2k + 1 for some integer k.
3. Therefore, n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1.
4. Since 2k² + 2k is an integer, n² is of the form 2m + 1 for some integer m.
5. Therefore, n² is odd.

2.2. Proofs by Contraposition: Flipping the Script

Proof by contraposition is a technique used to prove a statement of the form P → Q by proving its contrapositive, ¬Q → ¬P. This works because P → Q and ¬Q → ¬P are logically equivalent.

Strategy:

1. Identify the statement: Determine the hypothesis (P) and the conclusion (Q).
2. Formulate the contrapositive: Negate both the hypothesis and the conclusion, and reverse their order.
3. Prove the contrapositive: Use a direct proof to show that ¬Q → ¬P.
4. Conclude: Since you have proven the contrapositive, you have also proven the original statement.

Example:

Prove: If n² is an even integer, then n is an even integer.

Proof:

1. The statement is P → Q, where P is “n² is even” and Q is “n is even.”
2. The contrapositive is ¬Q → ¬P, which is “If n is not even (i.e., n is odd), then n² is not even (i.e., n² is odd).”
3. Prove the contrapositive:
   • Assume n is odd.
   • By definition, n = 2k + 1 for some integer k.
   • Therefore, n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1.
   • Since 2k² + 2k is an integer, n² is of the form 2m + 1 for some integer m.
   • Therefore, n² is odd.
4. Conclusion: Since we have proven the contrapositive, we have proven that if n² is even, then n is even.

2.3. Proofs by Contradiction: Seeking the Absurd

Proof by contradiction involves assuming the negation of the statement you want to prove and then deriving a contradiction. This shows that the initial assumption must be false, and therefore the original statement must be true.

Strategy:

1. Identify the statement: Clearly understand what you want to prove.
2. Assume the negation: Assume that the statement is false.
3. Derive a contradiction: Use logical deductions to arrive at a statement that is clearly false or contradicts a known fact or assumption.
4. Conclude: Since the assumption leads to a contradiction, it must be false, and therefore the original statement must be true.

Example:

Prove: There are infinitely many prime numbers.

Proof:

1. Assume the negation: There are finitely many prime numbers.
2. Let p₁, p₂, ..., pₙ be the complete list of prime numbers.
3. Consider the number N = (p₁ * p₂ * ... * pₙ) + 1.
4. N is either prime or composite.
   • If N is prime, then we have found a prime number not in our list, which contradicts our assumption.
   • If N is composite, then it must be divisible by some prime number pᵢ in our list.
   • However, if we divide N by any pᵢ, we get a remainder of 1 (since N = (p₁ * p₂ * ... * pₙ) + 1). This means N is not divisible by any prime number in our list, which is a contradiction.
5. Conclusion: Since our assumption leads to a contradiction, there must be infinitely many prime numbers.

2.4. Proofs by Induction: Climbing the Ladder

Proof by induction is used to prove statements about natural numbers. It involves two steps: a base case and an inductive step.

Strategy:

1. Identify the statement: Determine the statement P(n) that you want to prove for all natural numbers n.
2. Base Case: Prove that P(0) (or P(1), depending on the problem) is true. This is the starting point of the induction.
3. Inductive Hypothesis: Assume that P(k) is true for some arbitrary natural number k. This is the assumption you will use to prove the next step.
4. Inductive Step: Prove that P(k+1) is true, assuming that P(k) is true. This shows that if the statement is true for some value k, it must also be true for the next value k+1.
5. Conclude: By the principle of mathematical induction, the statement P(n) is true for all natural numbers n.

Example:

Prove: The sum of the first n positive integers is n(n+1)/2. That is, 1 + 2 + … + n = n(n+1)/2.

Proof:

1. Let P(n) be the statement “1 + 2 + … + n = n(n+1)/2”.
2. Base Case (n=1): 1 = 1(1+1)/2 = 1. So P(1) is true.
3. Inductive Hypothesis: Assume that P(k) is true for some arbitrary natural number k. That is, assume that 1 + 2 + … + k = k(k+1)/2.
4. Inductive Step: We need to show that P(k+1) is true, assuming that P(k) is true. That is, we need to show that 1 + 2 + … + k + (k+1) = (k+1)(k+2)/2.
   • Starting with the left side: 1 + 2 + … + k + (k+1) = k(k+1)/2 + (k+1) (by the inductive hypothesis).
   • Simplifying: k(k+1)/2 + (k+1) = (k(k+1) + 2(k+1))/2 = (k+1)(k+2)/2.
   • This is the right side, so P(k+1) is true.
5. Conclusion: By the principle of mathematical induction, the statement 1 + 2 + … + n = n(n+1)/2 is true for all natural numbers n.

2.5. Proofs by Cases: Dividing and Conquering

Proof by cases involves dividing the possible scenarios into a finite number of cases and proving the statement for each case.

Strategy:

1. Identify the statement: Understand what you need to prove.
2. Identify the cases: Determine all possible scenarios that could occur. Make sure that the cases are mutually exclusive and exhaustive (i.e., they cover all possibilities).
3. Prove each case: For each case, use direct proof, contraposition, contradiction, or any other appropriate technique to show that the statement is true under that scenario.
4. Conclude: Since you have proven the statement for all possible cases, the statement is true.

Example:

Prove: For any integer n, n² + n is an even integer.

Proof:

1. We want to prove that n² + n is even for any integer n.
2. There are two cases: