equivalence relations in mathematics k 16
play

Equivalence relations in mathematics, K-16+ Art Duval Department of - PowerPoint PPT Presentation

Jul 19, 2023 •208 likes •1.07k views

Definitions and motivation Examples More theory Equivalence relations in mathematics, K-16+ Art Duval Department of Mathematical Sciences University of Texas at El Paso AMS Southeastern Sectional Meeting University of Louisville October 5,

  1. Definitions and motivation Examples More theory Equivalence relations in mathematics, K-16+ Art Duval Department of Mathematical Sciences University of Texas at El Paso AMS Southeastern Sectional Meeting University of Louisville October 5, 2013 Art Duval Equivalence relations in mathematics, K-16+

  2. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = Art Duval Equivalence relations in mathematics, K-16+

  3. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 Art Duval Equivalence relations in mathematics, K-16+

  4. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 = 13 15 Art Duval Equivalence relations in mathematics, K-16+

  5. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 = 13 15 We have to use 2 3 = 10 15 and 1 5 = 3 15 . Art Duval Equivalence relations in mathematics, K-16+

  6. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 15 + 3 10 15 = 13 15 We have to use 2 3 = 10 15 and 1 5 = 3 15 . Questions: Art Duval Equivalence relations in mathematics, K-16+

  7. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 = 13 15 We have to use 2 3 = 10 15 and 1 5 = 3 15 . Questions: ◮ If 2 3 and 10 15 are equal, why can we use one but not the other? Art Duval Equivalence relations in mathematics, K-16+

  8. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 = 13 15 We have to use 2 3 = 10 15 and 1 5 = 3 15 . Questions: ◮ If 2 3 and 10 15 are equal, why can we use one but not the other? ◮ Could we have used something else besides 10 15 ? Art Duval Equivalence relations in mathematics, K-16+

  9. Definitions and motivation Fractions Examples Geometry More theory Real life One reason fractions are hard 2 3 + 1 5 = 10 15 + 3 15 = 13 15 We have to use 2 3 = 10 15 and 1 5 = 3 15 . Questions: ◮ If 2 3 and 10 15 are equal, why can we use one but not the other? ◮ Could we have used something else besides 10 15 ? ◮ Would we use something else in another situation, or should we always use 10 15 ? Art Duval Equivalence relations in mathematics, K-16+

  10. Definitions and motivation Fractions Examples Geometry More theory Real life Equivalent fractions Definition: a b ∼ c d if they reduce to the same fraction ( ad = bc ). Art Duval Equivalence relations in mathematics, K-16+

  11. Definitions and motivation Fractions Examples Geometry More theory Real life Equivalent fractions Definition: a b ∼ c d if they reduce to the same fraction ( ad = bc ). It’s easy to check ∼ is an equivalence relation, Art Duval Equivalence relations in mathematics, K-16+

  12. Definitions and motivation Fractions Examples Geometry More theory Real life Equivalent fractions Definition: a b ∼ c d if they reduce to the same fraction ( ad = bc ). It’s easy to check ∼ is an equivalence relation, so we can partition fractions as follows: a b and c d are in the same part (“equivalence class”) if a b ∼ c d . 1 17 2 10 1 3 4 20 2 34 3 15 5 15 7 35 4 6 4 14 10 8 40 16 8 12 6 21 50 40 70 28 10 7 20 8 2 7 8 36 20 14 20 12 10 35 14 63 Art Duval Equivalence relations in mathematics, K-16+

  13. Definitions and motivation Fractions Examples Geometry More theory Real life Adding fractions (revisited) b ∼ c a if d e f ∼ g and h Art Duval Equivalence relations in mathematics, K-16+

  14. Definitions and motivation Fractions Examples Geometry More theory Real life Adding fractions (revisited) a b ∼ c if d f ∼ g e and h a b + e f ∼ c d + g then h Art Duval Equivalence relations in mathematics, K-16+

  15. Definitions and motivation Fractions Examples Geometry More theory Real life Adding fractions (revisited) a b ∼ c if d e f ∼ g and h a b + e f ∼ c d + g then h So, really we should say � 2 � � 1 � � 13 � + = 3 5 15 , because anything equivalent to 2 3 plus anything equivalent to 1 5 “equals” something equivalent to 13 15 . Art Duval Equivalence relations in mathematics, K-16+

  16. Definitions and motivation Fractions Examples Geometry More theory Real life Adding fractions (revisited) a b ∼ c if d e f ∼ g and h a b + e f ∼ c d + g then h So, really we should say � 2 � � 1 � � 13 � + = 3 5 15 , because anything equivalent to 2 3 plus anything equivalent to 1 5 “equals” something equivalent to 13 15 . ◮ But it’s hard to compute unless we pick the right representative. Art Duval Equivalence relations in mathematics, K-16+

  17. Definitions and motivation Fractions Examples Geometry More theory Real life Adding fractions (revisited) a b ∼ c if d f ∼ g e and h a b + e f ∼ c d + g then h So, really we should say � 2 � � 1 � � 13 � + = 3 5 15 , because anything equivalent to 2 3 plus anything equivalent to 1 5 “equals” something equivalent to 13 15 . ◮ But it’s hard to compute unless we pick the right representative. ◮ In other settings, we stick to the fraction in lowest terms, a distinguished representative. Art Duval Equivalence relations in mathematics, K-16+

  18. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: Art Duval Equivalence relations in mathematics, K-16+

  19. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: ◮ Similarity ◮ same “shape”, possibly different size ◮ can get via dilation, reflection, rotation, translation Art Duval Equivalence relations in mathematics, K-16+

  20. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: ◮ Similarity ◮ same “shape”, possibly different size ◮ can get via dilation, reflection, rotation, translation ◮ Congruence ◮ same “shape”, size ◮ can get via reflection, rotation, translation Art Duval Equivalence relations in mathematics, K-16+

  21. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: ◮ Similarity ◮ same “shape”, possibly different size ◮ can get via dilation, reflection, rotation, translation ◮ Congruence ◮ same “shape”, size ◮ can get via reflection, rotation, translation ◮ Same shape, size, chirality ◮ can get via rotation, translation Art Duval Equivalence relations in mathematics, K-16+

  22. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: ◮ Similarity ◮ same “shape”, possibly different size ◮ can get via dilation, reflection, rotation, translation ◮ Congruence ◮ same “shape”, size ◮ can get via reflection, rotation, translation ◮ Same shape, size, chirality ◮ can get via rotation, translation ◮ Same shape, size, chirality, orientation ◮ can get via translation Art Duval Equivalence relations in mathematics, K-16+

  23. Definitions and motivation Fractions Examples Geometry More theory Real life Similarity, congruence, etc. Some equivalence relations from geometry: ◮ Similarity ◮ same “shape”, possibly different size ◮ can get via dilation, reflection, rotation, translation ◮ Congruence ◮ same “shape”, size ◮ can get via reflection, rotation, translation ◮ Same shape, size, chirality ◮ can get via rotation, translation ◮ Same shape, size, chirality, orientation ◮ can get via translation ◮ Same shape, size, chirality, orientation, position ◮ equality Art Duval Equivalence relations in mathematics, K-16+

  24. Definitions and motivation Fractions Examples Geometry More theory Real life Finer partitions Art Duval Equivalence relations in mathematics, K-16+

  25. Definitions and motivation Fractions Examples Geometry More theory Real life Finer partitions ◮ As we go down that ladder, we refine the partition, by splitting each part into more parts. Art Duval Equivalence relations in mathematics, K-16+

  26. Definitions and motivation Fractions Examples Geometry More theory Real life Finer partitions ◮ As we go down that ladder, we refine the partition, by splitting each part into more parts. ◮ Different situations call for different interpretations of when two shapes are “the same”. Art Duval Equivalence relations in mathematics, K-16+

  27. Definitions and motivation Fractions Examples Geometry More theory Real life Money ◮ At the store, 1 dollar equals 4 quarters equals 10 dimes. Art Duval Equivalence relations in mathematics, K-16+

  28. Definitions and motivation Fractions Examples Geometry More theory Real life Money ◮ At the store, 1 dollar equals 4 quarters equals 10 dimes. ◮ At old vending machines, dollar bad, coins good. Art Duval Equivalence relations in mathematics, K-16+

Recommend

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

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

Equivalence Relations and Subgroups Toby Kenney with R. Par and R. Wood Mathematics, Dalhousie University, Halifax, Canada CT07 2007JN1723 Toby Kenney with R. Par and R. Wood Equivalence Relations and Subgroups Quantales Recall that

406 views • 21 slides
Discrete Mathematics in Computer Science Equivalence Relations and Partitions Malte Helmert,

Discrete Mathematics in Computer Science Equivalence Relations and Partitions Malte Helmert,

Discrete Mathematics in Computer Science Equivalence Relations and Partitions Malte Helmert, Gabriele R oger University of Basel Relations: Recap A relation over sets S 1 , . . . , S n is a set R S 1 S n . Possible

1.25k views • 88 slides
Equivalence Relations {( a , b ) | a and b are from the the same state}. Observe that these

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

Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide01.html Equivalence Relations http://localhost/~senning/courses/ma229/slides/equivalence-relations/slide02.html Equivalence Relations prev | slides

235 views • 4 slides
Forcing, Equivalence Relations and Marker Structures S. Jackson (joint with S. Gao, E. Krohne,

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

Forcing, Equivalence Relations and Marker Structures S. Jackson (joint with S. Gao, E. Krohne, and B. Seward) Department of Mathematics University of North Texas August, 2013 Chapman University S. Jackson Forcing, Equivalence Relations and

589 views • 32 slides
7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary rela- tion that is reflexive,

7.5 EQUIVALENCE RELATIONS def: An equivalence relation is a binary rela- tion that is reflexive,

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 )

312 views • 4 slides
Borel equivalence relations and the Laver forcing Michal Doucha Charles University in Prague and

Borel equivalence relations and the Laver forcing Michal Doucha Charles University in Prague and

Borel equivalence relations and the Laver forcing Michal Doucha Charles University in Prague and Institute of Mathematics, AS July 8th, 2012 Michal Doucha Borel equivalence relations and the Laver forcing Introduction Vladimir Kanovei, Marcin

952 views • 30 slides
Todays Expedition TED highlights Equivalence Relations Equivalence Classes and

Todays Expedition TED highlights Equivalence Relations Equivalence Classes and

Todays Expedition TED highlights Equivalence Relations Equivalence Classes and Partitions Boolean Algebra Boolean functions Sum of Products expansion Logic gates and circuits Sections 8.5, 11.1-11.3 in the text

541 views • 15 slides
Section 4.2: Equivalence Relations What is equality? What is equivalence? Equality is

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

Section 4.2: Equivalence Relations What is equality? What is equivalence? Equality is more basic, fundamental concept. Every element in a set is equal only to itself. The basic equality relation {(x,x) | x S} We tend to assume

893 views • 61 slides
Group Colorings and Shift Equivalence Relations S. Gao, S. Jackson*, B. Seward Department of

Group Colorings and Shift Equivalence Relations S. Gao, S. Jackson*, B. Seward Department of

Overview First Proof General Proof Group Colorings and Shift Equivalence Relations S. Gao, S. Jackson*, B. Seward Department of Mathematics University of North Texas AMS-ASL Special Session on Logic and Dynamical Systems January, 2009

467 views • 27 slides
THE KNUTH RELATIONS HUAN VO Abstract. We define three equivalence relations on the set of words

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

THE KNUTH RELATIONS HUAN VO Abstract. We define three equivalence relations on the set of words using different means. It turns out that they are the same relation. 1. Motivation Recall the algorithm of row inserting an entry x into a tableau T .

239 views • 10 slides
Affine Toric Equivalence Relations are Effective Claudiu Raicu University of California, Berkeley

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

Affine Toric Equivalence Relations are Effective Claudiu Raicu University of California, Berkeley AMS-SMM Joint Meeting, Berkeley, June 2010 Motivating Question Under what circumstances do quotients by finite equivalence relations exist?

879 views • 39 slides
A decision procedure for equivalence relations Sbastien Michelland with Pierre Corbineau,

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

Congruence closure Equivalence and inclusions Quantified hypotheses Conclusion A decision procedure for equivalence relations Sbastien Michelland with Pierre Corbineau, Lionel Rieg and Karine Altisen July 5, 2020 1 / 14 (CC BY-ND)

660 views • 22 slides
Polishable Borel equivalence relations S lawomir Solecki Cornell University Research

Polishable Borel equivalence relations S lawomir Solecki Cornell University Research

Polishable Borel equivalence relations S lawomir Solecki Cornell University Research supported by NSF grant DMS1700426 June 2018 Outline Outline of Topics Introduction 1 Polishable equivalence relations 2 Canonical approximations

415 views • 28 slides
Discrete Mathematics in Computer Science B6. Equivalence and Order Relations Malte Helmert,

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

Discrete Mathematics in Computer Science B6. Equivalence and Order Relations Malte Helmert, Gabriele R oger University of Basel October 12, 2020 Malte Helmert, Gabriele R oger (University of Basel) Discrete Mathematics in Computer

612 views • 36 slides
Equivalence uandvarestrongly Relations connected. u G * * v AND v G * * u AND

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

MathematicsforComputerScience two-waywalks MIT 6.042J/18.062J walkfromutovand backfromvtou: Equivalence uandvarestrongly Relations connected. u G * * v AND v G * * u AND AlbertRMeyer

81 views • 5 slides
Equality and equivalence relations in formal proofs Pierre CORBINEAU DCS day, Autrans, 26-27

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

A short introduction to myself Equality and equivalence relations in Coq Equality and equivalence relations in formal proofs Pierre CORBINEAU DCS day, Autrans, 26-27 march 2009 A short introduction to myself Equality and equivalence relations

647 views • 25 slides
CSCI 246 Class 16 MORE EQUIVALENCE RELATIONS Quiz Questions Lecture 27: What is the

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

CSCI 246 Class 16 MORE EQUIVALENCE RELATIONS Quiz Questions Lecture 27: What is the difference between an equivalence relation and a partial order relation? Notes and Clarifications Extra credit due tonight Quiz option for

295 views • 25 slides
Countable Borel equivalence relations, recursion theory, and Borel combinatorics Andrew Marks UC

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

Countable Borel equivalence relations, recursion theory, and Borel combinatorics Andrew Marks UC Berkeley Countable Borel equivalence relations Definition A Borel equivalence relation E is an equivalence relation on 2 that has a 0

567 views • 56 slides
Borel equivalence relations and symmetric models Topologies for the Friedman-Stanley jumps Assaf

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

Borel equivalence relations and symmetric models Topologies for the Friedman-Stanley jumps Assaf Shani UCLA Logic colloquium, Udine, Italy July 2018 1 / 7 Motivation Given an equivalence relation E on a Polish space X , how does E behave

225 views • 7 slides
B5.1 Relations relation for natural numbers These are examples of binary relations,

B5.1 Relations relation for natural numbers These are examples of binary relations,

Discrete Mathematics in Computer Science October 7, 2020 B5. Relations Discrete Mathematics in Computer Science B5. Relations B5.1 Relations Malte Helmert, Gabriele R oger B5.2 Properties of Binary Relations University of Basel

532 views • 4 slides
Thin Projective Equivalence Relations and Inner Models Philipp Schlicht University of Mnster,

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

Thin Projective Equivalence Relations and Inner Models Philipp Schlicht University of Mnster, Germany Logic Colloquium Wrocaw, 14-19 July 2007 Definition. An equivalence relation E is called thin if there is no perfect set

279 views • 13 slides
Borel equivalence relations and symmetric models Assaf Shani CMU, Harvard Special session on

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

Borel equivalence relations and symmetric models Assaf Shani CMU, Harvard Special session on choiceless set theory and related areas Denver, January 2020 1 / 8 Friedman-Stanley jumps Definition Let E be an equivalence relation on X . A

209 views • 8 slides
Lecture 4.2: Equivalence relations and equivalence classes Matthew Macauley Department of

Lecture 4.2: Equivalence relations and equivalence classes Matthew Macauley Department of

Lecture 4.2: Equivalence relations and equivalence classes Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4190, Discrete Mathematical Structures M. Macauley (Clemson) Lecture

561 views • 11 slides
Adding two equivalence relations to the interval temporal logic AB Angelo Montanari 1 , Marco

Adding two equivalence relations to the interval temporal logic AB Angelo Montanari 1 , Marco

Adding two equivalence relations to the interval temporal logic AB Angelo Montanari 1 , Marco Pazzaglia 1 and Pietro Sala 2 1 University of Udine Department of Mathematics and Computer Science 2 University of Verona Department of Computer Science

710 views • 48 slides

More recommend