Mathematics I - Course Book Summary
This course book from IU International University of Applied Sciences covers fundamental math concepts essential for computer science, including number systems, logic, sets, and cryptography applications. You'll learn to understand mathematical texts, solve problems, and grasp the basics of cryptography.
What you'll get from this book:
- A solid understanding of basic mathematical principles.
- Skills to work with different number systems.
- Knowledge of logical statements and proof techniques.
- An introduction to applying math in cryptography.
Core Content:
1. Basic Mathematical Concepts:
- Number Sets: Learn the differences between natural (ℕ), whole (ℕ₀), integer (ℤ), rational (ℚ), and real (ℝ) numbers.
- Variables: Understand how variables are used as placeholders and how to define their value ranges (e.g., x ∈ ℕ).
- Equality Symbols: Distinguish between =, ≠, and := for equality, inequality, and definition, respectively.
2. Proof Techniques:
- Direct Proof: Start with prerequisites (A) and logically derive the assertion (B), showing A → B.
- Proof by Contradiction: Assume the opposite of what you want to prove and show that it leads to a contradiction.
- Proof by Contraposition: Instead of proving A → B, prove ¬B → ¬A.
- Proof by Induction: Prove a statement for the base case and then show that if it holds for n, it also holds for n+1.
3. Set Theory:
- Basic Definitions: Understand sets, subsets (N ⊆ M), supersets, and the empty set (∅).
- Set Operations: Learn to perform unions (M ∪ N), intersections (M ∩ N), and differences (M \ N) of sets.
- Important Rules: Grasp associative, distributive laws, and de Morgan's rules for set operations.
- Power Set: Know how the power set of a set is formed.
4. Propositional Logic:
- Logical Statements: Understand statements (true or false) and their logical equivalence (A ≡ B).
- Logical Operators: Use AND (∧), OR (∨), NOT (¬), implication (⇒), and equivalence (⇔) to form expressions.
- Truth Tables: Construct truth tables to determine the truth values of logical expressions.
- Simplification: Learn to simplify expressions using calculation rules and logical equivalences.
5. Number Systems:
- Decimal System: Understand base-ten representation.
- Binary System: Learn about bits, bytes, and binary arithmetic (addition, subtraction, multiplication).
- Two's Complement: Know how negative binary numbers are represented using two’s complement.
- Hexadecimal System: Work with base-16 representation and convert between binary and hexadecimal.
6. Functions:
- Basic Concepts: Define functions, domains, ranges, images, and preimages.
- Function Graphing: Create and interpret graphs of functions.
- Function Composition: Compose functions and understand the associative property.
- Special Features: Injective, surjective and bijective functions.
- Inverse Functions: Invertible functions and inverse function calculation.
7. Basic Algebraic Structures:
- Semigroup: A non-empty set and an associative operation.
- Identity Element: The definition and features of identity element.
- **Group:**A semigroup with a neutral element, in which each element is invertible.
- Ring: A non-empty set with two operations + and · with the properties.
- Residue Class Rings What they are and how to calculate in them.
8. Prime Numbers and Modular Arithmetic:
- Prime Numbers: Define prime numbers and understand the fundamental theorem of arithmetic.
- Euclid's Theorem: There are more prime numbers than there are numbers.
- Euclidean Algorithm: Use Euclidean Algorithm to find a greatest common divisor.
- Extended Euclidean Algorithm: Apply Extended Euclidean Algorithm to get a result.
- Primality Tests: Know different kinds of Primality Test.
9. Cryptography:
- Caesar Cipher: Encode and decode messages using the Caesar cipher.
- Symmetric vs. Asymmetric Cryptosystems: Weigh the pros and cons of symmetric and asymmetric cryptography.
- RSA Cryptosystem: Understand key generation, encryption, and decryption in RSA.
Q&A
Q: What is the difference between a subset and a superset?
A: If every element of set N is also in set M, then N is a subset of M (N ⊆ M), and M is a superset of N.
Q: What is the purpose of a truth table?
A: Truth tables are used to systematically determine the truth values of propositional logic formulas by considering all possible combinations of truth values for the variables.
Q: Why is the binary system important in computer science?
A: Computers operate using bits (binary digits), so the binary system is fundamental to how they store and process information.
Q: What is the key advantage of public-key cryptography?
A: Public-key (asymmetric) cryptography eliminates the need to exchange a secret key beforehand, making communication more secure.
Q: What is the purpose of the Euclidean algorithm?
A: The Euclidean algorithm efficiently finds the greatest common divisor (GCD) of two integers.
