Binary Relations (4B) Young Won Lim 5/15/17

Transcription

1 Binry Reltions (4B)

2 Copyright (c) Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny lter version published by the Free Softwre Foundtion; with no Invrint Sections, no Front-Cover Texts, nd no Bck-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documenttion License". Plese send corrections (or suggestions) to youngwlim@hotmil.com. This document ws produced by using OpenOffice nd Octve.

3 Crtesin Product Crtesin product A B of the sets A = { x, y, z } nd B = {, 2, } Reltions (4B)

4 Crtesin Coordintes Crtesin coordintes of exmple points Reltions (4B) 4

5 Reflexive Reltion Exmples Reltions (4B) 5

6 Symmetric Reltion Exmples Reltions (4B) 6

7 Anti-Symmetric Reltion Exmples Reltions (4B) 7

8 Trnsitive Reltion Exmples Reltions (4B) 8

9 Reflexive Reltion x (x, x) R Reltions (4B) 9

10 Symmetric Reltion x, y [ (x, y) R ( y, x) R ] symmetric symmetric symmetric symmetric symmetric symmetric Reltions (4B) 0

11 Not Symmetric Reltion { x, y [ (x, y) R ] [ ( y, x) R ] } x, y { [ (x, y) R ] [ ( y, x) R ] } x, y { [ (x, y) R ] [ ( y, x) R ] } x, y [ (x, y) R ] [ ( y, x) R ] x, y [ (x, y) R ] [ ( y, x) R ] counter exmple not symmetric not symmetric not symmetric Reltions (4B)

12 Anti-symmetric Reltion x, y [( (x, y) R ( y, x) R ) x = y ] nti-symmetric nti-symmetric nti-symmetric not nti-symmetric nti-symmetric nti-symmetric Reltions (4B) 2

13 Not Anti-symmetric Reltion x, y [ (x, y) R ( y, x) R ] [ x = y ] x, y { [ (x, y) R ( y, x) R ] [ x = y ]} x, y { [ (x, y) R ( y, x) R ] [ x = y ]} x, y { [ (x, y) R ( y, x) R ] [ x = y ]} x, y { [ (x, y) R ( y, x) R ] [ x y ]} counter exmple not nti-symmetric not nti-symmetric not nti-symmetric Reltions (4B)

14 Equivlent Anti-symmetric Reltion x, y [ (x, y) R ( y, x) R ] [ x = y ] x, y [ x y ] [ (x, y) R ( y, x) R ] No symmetric reltion is llowed neither symmetric nor nti-symmetric Reltions (4B) 4

15 Reflexive, Symmetric, Anti-symmetric x (x, x) R x, y [ (x, y) R ] [ ( y, x) R ] x, y [ (x, y) R ( y, x) R ] [ x = y ] Reflexive Also, symmetric (no reltion for (x, y) where x y) Also, nti-symmetric (no reltion for (x, y) where x y) not symmetric not nti-symmetric nti-symmetric symmetric Reltions (4B) 5

16 Reltion Exmples R {(,), (2, b), (,), (, b)} R 2 {(, x), (, y), (b, y), (b, z)} 2 b x y z Reltions (4B) 6

17 Composite Reltion Exmples R {(, ), (2, b), (, ), (,b)} R 2 {(, x), (, y), (b, y), (b, z)} 2 b x y z 2 x y z R 2 R {(, x), (, y), (2, y), (2, z), (, x), (, y), (, z)} Reltions (4B) 7

18 Composite Reltion Exmples R {(, ), (2, b), (, ), (,b)} R 2 {(, x), (, y), (b, y), (b, z)} R 2 R {(, x), (, y), (2, y), (2, z), (, x), (, y), (, z)} 2 b x y z 2 x y z b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 x y z ] 2 [ 0 ][ A A 2 = 0 0 ] [ 0 0 = 2 0 Reltions (4B) 8

19 Mtrix of Reltion R {(, ), (2, b), (, ), (,b)} R 2 {(, x), (, y), (b, y), (b, z)} b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 R 2 R {(, x), (, y), (2, y), (2, z), (, x), (, y), (, z)} x y z ] 2 [ 0 ][ A A 2 = 0 0 ] [ 0 0 = 2 0 Reltions (4B) 9

20 Composite Reltion Properties R {(, ), (2, b), (, ), (,b)} R 2 {(, x), (, y), (b, y), (b, z)} R 2 R {(, x), (, y), (2, y), (2, z), (, x), (, y), (, z)} b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 x y z ] 2 [ 0 ][ A A 2 = 0 0 ] [ 0 0 = 2 0 (i, k) R 2 R ik of A A x y z Reltions (4B) 20

21 Sufficient Prt [ 0 ][ A A 2 = 0 0 ] [ 0 = 0 0 j 2 ] j b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 i s b t b u v i su+tv su = tv = i {, 2, } j {x, y, z} s {0, } t {0, } su+tv 0 (s = ) (u = ) (t = ) (v = ) u {0, } v {0, } (i,) R (, j) R 2 (i, b) R (b, j) R 2 (i, j) R 2 R (i, j) R 2 R Reltions (4B) 2

22 Necessry Prt [ 0 ][ A A 2 = 0 0 ] [ 0 = 0 0 j 2 ] j b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 i s b t b u v i su+tv i {, 2, } j {x, y, z} s {0, } t {0, } u {0, } v {0, } (i, j) R 2 R (i,) R (, j) R 2 (i, b) R (b, j) R 2 (s = ) (u = ) su = (t = ) (v = ) tv = su+tv 0 Reltions (4B) 22

23 Necessry Prt [ 0 ][ A A 2 = 0 0 ] [ 0 = 0 0 j 2 ] j b [ 0 A = 2 0 ] x y z [ A 2 = 0 ] b 0 i s b t b u v i su+tv i {, 2, } j {x, y, z} s {0, } t {0, } u {0, } v {0, } su+tv 0 nonzero(i, j) th element of A A 2 (i, j) R 2 R Reltions (4B) 2

24 Trnsitivity Test Exmples R {(,), (b,b), (c,c), (d,d), (b, c), (c, b)} R R {(,), (b,b), (c,c), (d,d), (b,c), (c, b)} A = b c d b c d [ ] A A = [ ][ ] 0 0 = b c d b c d [ ] Reltions (4B) 24

25 Trnsitivity Test A = 2 4 [ 2 4 ] A 2 = 2 4 [ ] [ e ] b c d 2 f g 4 h = 2 4 [ 2 4 ] nonzero (i, j) th element of A 2 nonzero (i, j) th element of A e = (2,) R (,2) R b f = c g = d h = (2,2) R (2,2) R (2,) R (,2) R (2, 4) R (4,2) R (2,2) R Reltions (4B) 25

26 Binry Reltions nd Digrphs A = {0,,2,,4, 5,6} R A A R = {(,b) b (mod )} R = [ ] Reltions (4B) 26

27 Reflexive Reltion A = {0,,2,,4, 5,6} R A A R = {(,b) b (mod )} R = [ ] Reltions (4B) 27

28 Symmetric Reltion A = {0,,2,,4, 5,6} R A A R = {(,b) b (mod )} R = [ ] Reltions (4B) 28

29 Trnsitive Reltion A = {0,,2,,4, 5,6} 0 R A A R = {(,b) b (mod )} RR = [ ] Reltions (4B) 29

30 Equivlence Reltion A = {0,,2,,4, 5,6} R A A R = {(,b) b (mod )} Equivlence Reltion Reflexive Reltion & Symmetric Reltion & Trnsitive Reltion Reltions (4B) 0

31 Equivlence Clss A = Z + = {0,,2,, 4,5, 6, } R A A R = {(,b) b (mod )} {0,, 6, 9, } [0] [] {, 4, 7, 0, } [] [] {2, 5, 8,, } [2] [2] Reltions (4B)

32 References [] [2]