Prove that the relation of congruency is an equivalence relation on the set Z of all integers.

Asked by abhinavsaini123 | 29th May, 2015, 07:40: PM

Expert Answer:

T h e space r e l a t i o n space R space o f space c o n g r u e n c y space o n space t h e space s e t space o f space i n t e g e r s space i s space d e f i n e d space a s space open parentheses a comma b close parentheses element of R left right double arrow a identical to b space left parenthesis m o d space n right parenthesis comma space w h i c h space i m p l i e s a minus b space i s space d i v i s i b l e space b y space apostrophe n apostrophe.  R e f l e x i v e colon space F o r space apostrophe a apostrophe element of Z comma space a minus a equals 0 comma space w h i c h space i s space d i v i s i b l e space b y space apostrophe n apostrophe. space H e n c e space a identical to a space left parenthesis m o d space n right parenthesis space rightwards double arrow open parentheses a comma a close parentheses element of R H e n c e comma space t h e space r e l a t i o n space i s space R e f l e x i v e.  S y m m e t r i c colon space I f space open parentheses a comma b close parentheses element of R rightwards double arrow a identical to b space left parenthesis m o d space n right parenthesis left right double arrow b identical to a space left parenthesis m o d space n right parenthesis rightwards double arrow open parentheses b comma a close parentheses element of R H e n c e comma space open parentheses a comma b close parentheses element of R rightwards double arrow open parentheses b comma a close parentheses element of R. space H e n c e comma space t h e space g i v e n space r e l a t i o n space i s space S y m m e t r i c.  T r a n s i t i v e colon space I f space open parentheses a comma b close parentheses element of R rightwards double arrow a identical to b space left parenthesis m o d space n right parenthesis rightwards double arrow a minus b space i s space d i v i s i b l e space b y space apostrophe n apostrophe rightwards double arrow a minus b equals P n.. left parenthesis 1 right parenthesis I f space open parentheses b comma c close parentheses element of R rightwards double arrow b identical to c space left parenthesis m o d space n right parenthesis rightwards double arrow b minus c space i s space d i v i s i b l e space b y space apostrophe n apostrophe rightwards double arrow b minus c equals Q n.. left parenthesis 2 right parenthesis A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space g e t space a minus c equals open parentheses P plus Q close parentheses n rightwards double arrow a minus c space i s space d i v i s i b l e space b y space apostrophe n apostrophe rightwards double arrow a identical to c space left parenthesis m o d space n right parenthesis rightwards double arrow open parentheses a comma c close parentheses element of R H e n c e comma space t h e space r e l a t i o n space i s space t r a n s i t i v e. S i n c e comma space t h e space r e l a t i o n space i s space r e f l e x i v e comma space s y m m e t r i c space a n d space t r a n s i t i v e comma space i t space i s space a n space e q u i v a l e n c e space r e l a t i o n.

Answered by satyajit samal | 31st May, 2015, 02:06: PM

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