Book contents
- Frontmatter
- Contents
- Preface
- On the Structure of Mathematics
- Brief Summaries of Topics
- 1 Linear Algebra
- 2 ∈ and δ Real Analysis
- 3 Calculus for Vector-Valued Functions
- 4 Point Set Topology
- 5 Classical Stokes' Theorems
- 6 Differential Forms and Stokes' Thm.
- 7 Curvature for Curves and Surfaces
- 8 Geometry
- 9 Complex Analysis
- 10 Countability and the Axiom of Choice
- 11 Algebra
- 12 Lebesgue Integration
- 13 Fourier Analysis
- 14 Differential Equations
- 15 Combinatorics and Probability
- 16 Algorithms
- A Equivalence Relations
- Bibliography
- Index
A - Equivalence Relations
Published online by Cambridge University Press: 11 April 2011
- Frontmatter
- Contents
- Preface
- On the Structure of Mathematics
- Brief Summaries of Topics
- 1 Linear Algebra
- 2 ∈ and δ Real Analysis
- 3 Calculus for Vector-Valued Functions
- 4 Point Set Topology
- 5 Classical Stokes' Theorems
- 6 Differential Forms and Stokes' Thm.
- 7 Curvature for Curves and Surfaces
- 8 Geometry
- 9 Complex Analysis
- 10 Countability and the Axiom of Choice
- 11 Algebra
- 12 Lebesgue Integration
- 13 Fourier Analysis
- 14 Differential Equations
- 15 Combinatorics and Probability
- 16 Algorithms
- A Equivalence Relations
- Bibliography
- Index
Summary
Throughout this text we have used equivalence relations. Here we collect some of the basic facts about equivalence relations. In essence, an equivalence relation is a generalization of equality.
Definition A.0.1 (Equivalence Relation)An equivalence relation on a set X is any relation ‘x ∼ y’ for x, y ∈ X such that
1. (Reflexivity) For any x ∈ X, we have x ∼ x.
2. (Symmetry) For all x, y ∈ X, if x ∼ y then y ∼ x.
3. (Transitivity) For all x, y, z ∈ X, if x ∼ y and y ∼ z, then x ∼ z.
The basic example is that of equality. Another example would be when X = R and we say that x ∼ y if x − y is an integer. On the other hand, the relation x ∼ y if x ≤ y is not an equivalence relation, as it is not symmetric.
We can also define equivalence relations in term of subsets of the ordered pairs X × X as follows:
Definition A.0.2 (Equivalence Relation)An equivalence relation on a set X is a subset R ⊂ X × X such that
1. (Reflexivity) For any x ∈ X, we have (x, x) ∈ R.
2. (Symmetry) For all x, y ∈ X, if (x, y) ∈ R then (y,x) ∈ R.
3. (Transitivity) For all x, y, z ∈ X, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
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- All the Mathematics You MissedBut Need to Know for Graduate School, pp. 327 - 328Publisher: Cambridge University PressPrint publication year: 2001