Abstract mathematics, often simply referred to as abstract algebra or simply abstraction in mathematics, deals with concepts and structures that are generalized beyond the specific instances we often encounter in everyday mathematics. It focuses on finding patterns, creating frameworks, and understanding the relationships between different mathematical entities in a more theoretical way.
Here are a few key concepts in abstract mathematics, explained with examples:
1. Sets and Set Theory
A set is a collection of distinct objects, considered as an object in its own right. The study of sets allows for more abstract reasoning about collections of items.
Example: The set of natural numbers, \( \mathbb{N} = {0, 1, 2, 3, \ldots} \), can be studied using various operations, and one may define functions, relations, and more abstract structures over these sets.
2. Groups
A group is a set equipped with a binary operation that satisfies four properties: closure, associativity, the existence of an identity element, and the existence of inverse elements.
Example: The set of integers \( \mathbb{Z} \) under addition forms a group. Closure (the sum of any two integers is an integer), associativity (the sum does not depend on how the terms are grouped), the identity element (0, as adding zero does not change the integer), and the existence of inverses (for any integer \( n \), there exists \( -n \) such that \( n + (-n) = 0 \)) are all satisfied.
3. Rings
A ring is a set equipped with two binary operations, typically addition and multiplication, where addition forms an abelian group and multiplication is associative, and multiplication distributes over addition.
Example: The set of integers \( \mathbb{Z} \) with the usual addition and multiplication forms a ring.
4. Fields
A field is a set in which addition, subtraction, multiplication, and division (except by zero) are defined and satisfy the properties of commutativity, associativity, distributivity, and the existence of identity and inverses for both addition and multiplication.
Example: The set of rational numbers \( \mathbb{Q} \) forms a field. It allows for addition, subtraction, multiplication, and division (except by zero) while satisfying the properties necessary for a field.
5. Vector Spaces
A vector space is a collection of vectors that can be scaled and added together in a way that satisfies certain axioms, such as associativity, commutativity, and the existence of additive identities and inverses.
Example: The set of all ordered pairs of real numbers \( \mathbb{R}^2 \) can be viewed as a vector space over the field of real numbers \( \mathbb{R} \). You can add any two pairs and multiply by a real number, obeying the axioms required for vector spaces.
6. Topological Spaces
A topological space is a set endowed with a topology, which is a collection of open sets that satisfy certain axioms. It allows for the study of convergence, continuity, and compactness abstractly.
Example: The set of real numbers \( \mathbb{R} \) with the standard topology (where open sets are all unions of open intervals) is a common example used in analysis.
Conclusion
Abstract mathematics allows mathematicians to study the underlying structures and relationships that govern various kinds of mathematical objects. It emphasizes concepts that can apply across different contexts rather than being limited to specific numerical calculations or geometric shapes, thus generalizing and unifying numerous branches of mathematics. This abstraction is crucial for advanced topics such as algebra, analysis, topology, and beyond.