Relations (3A) Young Won Lim 3/15/18

Transcription

1 Reltions (A)

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. 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 LibreOffice nd Octve.

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

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

5 Crtesin Product 4 5 (,) (,) (,) (,4) (,5) (,) (,) (,) (,4) (,5) (,) (,) (,) (,4) (,5) 4 (4,) (4,) (4,) (4,4) (4,5) 5 (5,) (5,) (5,) (5,4) (5,5) Reltions (4B) 5

6 Crtesin Product (,) (,) (,) (,4) (,5) R R R R4 R5 (,) (,) (,) (,4) (,5) R R R R4 R5 (,) (,) (,) (,4) (,5) R R R R4 R5 4 (4,) (4,) (4,) (4,4) (4,5) 4 R 4R 4R 4R4 4R5 5 (5,) (5,) (5,) (5,4) (5,5) 5 5R 5R 5R 5R4 5R5 Reltions (4B) 6

7 Types of Reltions () x R x Reflexive Reltion Symmetric Reltion Trnsitive Reltion x R y y R x x R y y R z x R z Reltions (4B) 7

8 Types of Reltions () Reflexive: for ll x in X it holds tht xrx. Symmetric: for ll x nd y in X it holds tht if xry then yrx. Antisymmetric: for ll x nd y in X, if xry nd yrx then x = y. Trnsitive: for ll x, y nd z in X it holds tht if xry nd yrz then xrz. Reltions (4B) 8

9 Reltion Exmples () x y 4 5 x > y 4 5 (,) (,) (,) (,) (,) (,) (,) (,) (,) 4 (4,) (4,) (4,) (4,4) 4 (4,) (4,) (4,) 5 (5,) (5,) (5,) (5,4) (5,5) 5 (5,) (5,) (5,) (5,4) Reltions (4B) 9

10 Reltion Exmples () x = y x = y (,) (,) (,) (,) (,) 4 (4,4) 4 (4,) 5 (5,5) 5 (5,4) Reltions (4B) 0

11 Reltion Exmples () x+ y = 4 x+ y (,) (,) (,) (,) (,) (,) (,) (,) (,) Reltions (4B)

12 Reflexive Reltion Exmples Reflexive Reltion Irreflexive Reltion Reltions (4B)

13 Symmetric Reltion Exmples Symmetric Reltion Reltions (4B)

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

15 Trnsitive Reltion Exmples Reltions (4B) 5

16 Reflexive Reltion x (x, x) R Reltions (4B) 6

17 Symmetric Reltion x, y [ (x, y) R ( y, x) R ] symmetric symmetric symmetric symmetric symmetric symmetric Reltions (4B) 7

18 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) 8

19 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) 9

20 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) 0

21 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)

22 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)

23 Reltion Exmples R {(,), (, b), (,), (, b)} R {(, x), (, y), (b, y), (b, z)} b x y z Reltions (4B)

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

25 Composite Reltion Exmples R {(, ), (, b), (, ), (,b)} R {(, x), (, y), (b, y), (b, z)} R R {(, x), (, y), (, y), (, z), (, x), (, y), (, z)} b x y z x y z b [ 0 A = 0 ] x y z [ A = 0 ] b 0 x y z ] [ 0 ][ A A = 0 0 ] [ 0 0 = 0 Reltions (4B) 5

26 Mtrix of Reltion R {(, ), (, b), (, ), (,b)} R {(, x), (, y), (b, y), (b, z)} b [ 0 A = 0 ] x y z [ A = 0 ] b 0 R R {(, x), (, y), (, y), (, z), (, x), (, y), (, z)} x y z ] [ 0 ][ A A = 0 0 ] [ 0 0 = 0 Reltions (4B) 6

27 Composite Reltion Properties R {(, ), (, b), (, ), (,b)} R {(, x), (, y), (b, y), (b, z)} R R {(, x), (, y), (, y), (, z), (, x), (, y), (, z)} b [ 0 A = 0 ] x y z [ A = 0 ] b 0 x y z ] [ 0 ][ A A = 0 0 ] [ 0 0 = 0 (i, k) R R ik of A A 0 x y z Reltions (4B) 7

28 Sufficient Prt [ 0 ][ A A = 0 0 ] [ 0 = 0 0 j ] j b [ 0 A = 0 ] x y z [ A = 0 ] b 0 i s b t b u v i su+tv su = tv = i {,, } j {x, y, z} s {0, } t {0, } su+tv 0 (s = ) (u = ) (t = ) (v = ) u {0, } v {0, } (i,) R (, j) R (i, b) R (b, j) R (i, j) R R (i, j) R R Reltions (4B) 8

29 Necessry Prt [ 0 ][ A A = 0 0 ] [ 0 = 0 0 j ] j b [ 0 A = 0 ] x y z [ A = 0 ] b 0 i s b t b u v i su+tv i {,, } j {x, y, z} s {0, } t {0, } u {0, } v {0, } (i, j) R R (i,) R (, j) R (i, b) R (b, j) R (s = ) (u = ) su = (t = ) (v = ) tv = su+tv 0 Reltions (4B) 9

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

31 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)

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

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

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

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

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

37 Equivlence Reltion A = {0,,,,4, 5,6} R A A R = {(,b) b (mod )} Equivlence Reltion Reflexive Reltion & Symmetric Reltion & Trnsitive Reltion Reltions (4B) 7

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

39 References [] []