MAT102H5 Lecture 37: 11.29 Lec
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1h29 LEC
Examples :
①on IR R={ Lays :X
'-142=13 EIRXIR
a
I
J
- l I
-i
CD Not reflexive COso )#Ras O
-
to
'
El
•Is symmetric ,as if x4y2=1 ,then yatx I
.Not transitive co ,1) C- Rand Cl .ER but Cos 01 &R
②On 2:R-
-flab ):at4b is divisible by 5)
.Is reflexive ,as a-149=59 is divisible by 5 for any AEZ
Cat4bJtLbt4as=5at5b ⇒bt4a=c5at5b )-Catty
for any aob 82
Since status and 51 5atFb we get bTbt4q so Ris symmetric
If 5lat4b and 51 bt 4C (fer ab CEZ ).
then Ella
-14534lb -14C )⇒5lat5bt4C
5lcat5bt4c )-15 b) ⇒Flat 4C So Ris transitive
Ris an equivalent relation .
Remark :For equivalence equation ,we often use symbols such as I,
N.X,=
Eg :1=-11 instead of Cb IDEK
713 Equivalence Classes
Definition :Lee Rbe an equivalence relation on aset S,and xes ,We define the
equivalence class of xas the set .
[x]=fyes :C x. YSER)
Example :On IR :Xuy if and only if X,yhave the same integer part
Eg .The integer part of 3,14 is 3
- ' i ' -i i- -2,7 IS -2
The equivalence class of 3114 is 53,4 )
i. i - ' ' a i -2.7 is €3
,
⇒
0.3 is Gl ,D.
