7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary - PDF document

7.5.1 Section 7.5 Equivalence Relations 7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. Set S = { a, b, c, d, e, f } and Example 7.5.1: � R = ( a, a ) , ( b, c ) , ( c, b ) , ( d, e ) , � ( d, f ) , ( e, d ) , ( e, f ) , ( f, d ) , ( f, e ) a b c d e f a 1 d b 1 1 a c 1 1 d 1 1 1 e f e 1 1 1 b c f 1 1 1 Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed.

7.5.2 Chapter 7 RELATIONS Prop 7.5.1. Let R be an equivalence relation. Then every component of the digraph of R is a ♦ complete digraph. Cor 7.5.2. An equivalence relation induces a partition on its domain. The vertex set of each component of Proof: the digraph is a cell of the partition. ♦ Prop 7.5.3. Let R be an equivalence relation. Order its domain so that the elements of each cell of the induced partition occur contiguously. The resulting matrix represention has blocks of square matrices of all 1’s down its main diagonal and has all zeroes for its other entries. ♦ Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed.

7.5.3 Section 7.5 Equivalence Relations FINITE EQUIVALENCE RELATIONS 2 × 2 checkerboards Example 7.5.2: There are 16 checkerboards. Checkerboard x is related to checkerboard y if it can be trans- formed into y by a rotation or by a reflection. relation R = sibling of Example 7.5.3: domain = all persons The reflexive, trans. closure of R is an eq rel. It partitions all of humanity into equivalence classes of (full) siblings. Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed.

7.5.4 Chapter 7 RELATIONS INFINITE EQUIVALENCE RELATIONS domain = rational fractions Example 7.5.4: � p � � � p, q ∈ Z , q � = 0 � q a b and c Then are related if ad = bc d The partition cells are rational fractions of equal value. COUNTING PROBLEM (solved in w4205): How many cells are there if p, q ∈ { 1 , . . . , 10 } ? domain Z Example 7.5.5: eq. rel. = congruence mod 3 Equivalence Classes: [0] 3 = { . . . , − 6 , − 3 , 0 , 3 , 6 , 9 , . . . } [1] 3 = { . . . , − 5 , − 2 , 1 , 4 , 7 , 10 , . . . } [2] 3 = { . . . , − 4 , − 1 , 2 , 5 , 8 , 11 , . . . } domain = propositions on Example 7.5.6: p, q (infinite domain – arbitrarily long strings) eq. rel. = logical equivalence N.B. there are 2 4 cells to this partition Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed.

7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary - PDF document

7.5.1 Section 7.5 Equivalence Relations 7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary rela- tion that is reflexive, symmetric, and transitive. Set S = { a, b, c, d, e, f } and Example 7.5.1: R = ( a, a ) , ( b, c )

Equivalence Relations {( a , b ) | a and b are from the the same state}. Observe that these

Todays Expedition TED highlights Equivalence Relations Equivalence Classes and

Section 4.2: Equivalence Relations What is equality? What is equivalence? Equality is

THE KNUTH RELATIONS HUAN VO Abstract. We define three equivalence relations on the set of words

Affine Toric Equivalence Relations are Effective Claudiu Raicu University of California, Berkeley

A decision procedure for equivalence relations Sbastien Michelland with Pierre Corbineau,

Polishable Borel equivalence relations S lawomir Solecki Cornell University Research

Equivalence uandvarestrongly Relations connected. u G * * v AND v G * * u AND

Forcing, Equivalence Relations and Marker Structures S. Jackson (joint with S. Gao, E. Krohne,

Equality and equivalence relations in formal proofs Pierre CORBINEAU DCS day, Autrans, 26-27

CSCI 246 Class 16 MORE EQUIVALENCE RELATIONS Quiz Questions Lecture 27: What is the

Equivalence Relations and Subgroups Toby Kenney with R. Par and R. Wood Mathematics, Dalhousie

Countable Borel equivalence relations, recursion theory, and Borel combinatorics Andrew Marks UC

Borel equivalence relations and symmetric models Topologies for the Friedman-Stanley jumps Assaf

Thin Projective Equivalence Relations and Inner Models Philipp Schlicht University of Mnster,

Borel equivalence relations and symmetric models Assaf Shani CMU, Harvard Special session on

Organisatorisches Verification Technology gy Prof. Dr.-Ing. Hans Eveking C Computer Systems

! Quantum Computation WARNING I DO NOT KNOW ANYTHING ABOUT PHYSICS. SERIOUSLY. Image: Centre

ECE 3060 VLSI and Advanced Digital Design Lecture 6 Gate Delay and Logical Effort ECE 3060

A coinductive approach to -equivalence relations jww Simon Boulier and Nicolas Tabareau

A Type Theory with Partial Equivalence Relations as Types Abhishek Anand Mark Bickford Robert

Discrete Mathematics in Computer Science B6. Equivalence and Order Relations Malte Helmert,

CS156: The Calculus of Computation Zohar Manna Winter 2010 Chapter 9: Quantifier-free Equality

The quasi-isometry relation for finitely generated groups Simon Thomas Rutgers University 25th