🔶Intro to Abstract Math Unit 1 – Intro to Abstract Math

Key Concepts and Definitions

• Abstract mathematics focuses on the study of mathematical structures and their properties independent of their specific representations
• Axioms serve as the foundational statements or assumptions used to build mathematical theories and derive theorems
• Theorems are mathematical statements that can be proven using logical arguments based on axioms and previously established results
• Proofs provide rigorous demonstrations of the truth of mathematical statements using logical reasoning and deduction
• Mathematical logic involves the study of formal systems, including propositional and predicate logic, used to represent and analyze mathematical arguments
• Sets are fundamental objects in mathematics that represent collections of distinct elements (numbers, symbols, or other mathematical objects)
• Functions describe mappings between sets that assign a unique output value to each input value from the domain set
• Number theory explores the properties and relationships of integers (whole numbers) and their generalizations

Logic and Proof Techniques

• Propositional logic deals with statements (propositions) and logical connectives (and, or, not, implies) to form compound statements
  • Truth tables help determine the truth values of compound statements based on the truth values of their constituent propositions
• Predicate logic extends propositional logic by introducing quantifiers (for all, there exists) to express statements about elements of a set
• Logical equivalence refers to the property of two statements having the same truth value for all possible truth assignments of their variables
• Direct proofs establish the truth of a statement by starting with known facts and using logical deductions to reach the desired conclusion
• Proof by contradiction assumes the negation of the statement to be proved and derives a logical contradiction, thus proving the original statement
• Proof by induction is a technique used to prove statements involving natural numbers by proving a base case and an inductive step
  • The inductive step assumes the statement holds for a specific value $k$ and proves it holds for $k+1$

Set Theory Fundamentals

• Sets can be described by listing their elements within curly braces $\{a, b, c\}$ or using set-builder notation $\{x | P(x)\}$, where $P(x)$ is a predicate
• Subsets are sets whose elements are all contained within another set (A is a subset of B if every element of A is also an element of B)
• Power sets consist of all possible subsets of a given set, including the empty set and the set itself
• Set operations include union (elements in either set), intersection (elements common to both sets), and complement (elements not in the set)
• Cartesian products of two sets A and B, denoted $A \times B$, consist of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$
• Cardinality refers to the number of elements in a set, with finite sets having a natural number cardinality and infinite sets having transfinite cardinalities
  • Countable sets have a cardinality equal to or less than the cardinality of the natural numbers, while uncountable sets (real numbers) have a greater cardinality

Relations and Functions

• Relations between sets are subsets of their Cartesian product, representing connections or associations between elements
• Equivalence relations are reflexive (each element is related to itself), symmetric (if $a$ is related to $b$, then $b$ is related to $a$), and transitive (if $a$ is related to $b$ and $b$ is related to $c$, then $a$ is related to $c$)
  • Equivalence classes partition a set into disjoint subsets based on an equivalence relation
• Partial orders are reflexive, antisymmetric (if $a \leq b$ and $b \leq a$, then $a = b$), and transitive, representing a ordering of elements (real numbers with $\leq$)
• Functions map each element of the domain set to a unique element of the codomain set, with the set of all outputs forming the range
• Injective (one-to-one) functions map distinct inputs to distinct outputs, while surjective (onto) functions have their range equal to the codomain
• Bijective functions are both injective and surjective, establishing a one-to-one correspondence between the domain and codomain sets

Number Theory Basics

• Divisibility relates two integers $a$ and $b$, with $a$ dividing $b$ if there exists an integer $k$ such that $b = ak$
• Prime numbers have exactly two positive divisors, 1 and the number itself, while composite numbers have more than two divisors
• The fundamental theorem of arithmetic states that every positive integer greater than 1 can be uniquely expressed as a product of prime numbers
• Modular arithmetic involves performing arithmetic operations on integers within a fixed range, with numbers "wrapping around" after reaching the modulus
  • Congruence modulo $n$ states that two integers $a$ and $b$ are congruent (mod $n$) if their difference $a-b$ is divisible by $n$
• Greatest common divisors (GCD) and least common multiples (LCM) are important concepts in number theory for understanding the relationships between integers
• Diophantine equations are polynomial equations with integer coefficients for which integer solutions are sought (Pythagorean triples)

Introduction to Abstract Structures

• Groups are sets equipped with a binary operation satisfying closure, associativity, identity, and inverse properties (integers under addition, non-zero real numbers under multiplication)
  • Abelian groups are groups in which the binary operation is commutative
• Rings are sets with two binary operations, typically called addition and multiplication, satisfying specific axioms (integers, polynomials)
  • Commutative rings have a commutative multiplication operation
• Fields are commutative rings with unity in which every non-zero element has a multiplicative inverse (rational numbers, real numbers, complex numbers)
• Vector spaces are sets of objects (vectors) that can be added together and multiplied by scalars from a field, satisfying specific axioms
• Modules generalize vector spaces by allowing scalars from a ring instead of a field
• Algebras are vector spaces equipped with a bilinear operation called multiplication, generalizing the concept of rings (matrix algebras)

Problem-Solving Strategies

• Understand the problem by carefully reading the problem statement, identifying given information, and clarifying the desired outcome
• Break down complex problems into smaller, more manageable sub-problems that can be solved individually
• Look for patterns or similarities to previously encountered problems, as they may suggest potential solution approaches
• Consider multiple representations of the problem, such as algebraic, graphical, or numerical representations, to gain different insights
• Work backwards from the desired outcome to determine the necessary steps or conditions required to reach the solution
• Employ logical reasoning and proof techniques to validate proposed solutions and ensure their correctness
• Generalize solutions to broader classes of problems by identifying key characteristics and abstracting away specific details

Real-World Applications

• Cryptography utilizes number theory and abstract algebra concepts to develop secure communication and data protection systems (RSA encryption)
• Error-correcting codes, based on algebraic structures, enable reliable data transmission and storage by detecting and correcting errors (Reed-Solomon codes)
• Optimization problems in various fields, such as engineering and economics, rely on mathematical modeling and problem-solving techniques
• Computer science heavily relies on abstract mathematics for algorithm design, data structures, and the analysis of computational complexity
• Mathematical physics employs abstract structures to formulate and analyze physical theories (Hilbert spaces in quantum mechanics)
• Theoretical computer science uses logic, set theory, and abstract structures to study the foundations of computation and computability
• Abstract mathematics provides a rigorous foundation for various branches of mathematics, enabling the development of new theories and the discovery of underlying connections