कक्षा 6–8Foundationsनियोजित
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
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
2 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- 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.
आगे ले जाता है
वैकल्पिक खोजें
- 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.विस्मय पथ →
- 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.विस्मय पथ →
- 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
अधिगम पथ
आगे पढ़ने के लिए
- Daniel J. Velleman, How to Prove It: A Structured Approach (1994)