六至八年级Foundations计划中

Relations, equivalence relations and partitions

Relations as sets of ordered pairs, and the special relations (reflexive, symmetric and transitive) that sort a set into non-overlapping classes. 'Same remainder', 'same colour' and 'same shape' are first examples.

math.found.equivalence-relations

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前置知识

Sets

2 项必需的前置知识（一直追溯到底）

你将学会

Represent a relation on a finite set as a set of ordered pairs, a table and an arrow diagram.
Test a relation for reflexivity, symmetry and transitivity, giving a counterexample whenever a property fails.
Pass between an equivalence relation and the partition formed by its equivalence classes.

定义

A relation R on a set A is a subset of A × A; we write a R b when (a, b) ∈ R.
R is an equivalence relation if it is reflexive (a R a), symmetric (a R b implies b R a) and transitive (a R b and b R c imply a R c).
The equivalence class of a is [a] = {x ∈ A : x R a}.
A partition of A is a set of non-empty, pairwise disjoint subsets of A whose union is A.

定理

The equivalence classes of an equivalence relation on A form a partition of A, and every partition of A comes from exactly one equivalence relation.
Any two equivalence classes are either equal or disjoint.
For n ≥ 1, 'has the same remainder on division by n' is an equivalence relation on ℤ with exactly n classes.

常见误解

Thinking symmetry and transitivity together imply reflexivity; the argument fails for an element related to nothing.
Treating an equivalence class as a single element instead of a set of elements.
Assuming every relation is symmetric, as 'is less than' shows it need not be.

通向

Orders, lattices and well-orders Congruences Constructing ℤ, ℚ and ℝ Groups

可选发现

Quotient groups and quotient rings Collapsing each class to a single point gives a quotient set; when the relation respects an operation, the quotient inherits it, giving quotient groups and rings such as ℤ/nℤ.探奇小径 →
Universal properties The quotient map A → A/~ is universal: every function out of A that is constant on equivalence classes factors through it in exactly one way.探奇小径 →
Metric spaces and point-set topology Gluing points by an equivalence relation produces quotient spaces, such as a circle from an interval with its endpoints identified.
← Collections
← Integers

学习路径

Accelerated secondary progression