# Show that the relation R = {(a, b) : a, bZ and a+b is even} is an equivalence relation on the set of integers.

# Let S be a relation on the set R of all real numbers defined by

S = {(a, b)**R*R** : }

Prove that S is not an equivalence relation on R.

Asked by abhinavsaini123 | 29th May, 2015, 07:52: PM

Expert Answer:

Please ask one query in a single post. The explanation for the first question is given below:

$T h e space r e a l t i o n space left curly bracket left parenthesis a comma b right parenthesis colon space a comma b element of Z space a n d space a plus b space i s space e v e n right curly bracket. R e f l e x i v i t y colon F o r space a element of Z comma space a plus a equals 2 a space i s space a n space e v e n space n u m b e r. space H e n c e comma space open parentheses a comma a close parentheses element of R space f o r space a l l space a element of Z H e n c e comma space t h e space r e l a t i o n space i s space r e f l e x i v e. S y m m e t r i c colon I f space open parentheses a comma b close parentheses element of R rightwards double arrow a plus b space i s space e v e n rightwards double arrow a plus b equals 2 M rightwards double arrow b plus a equals 2 M rightwards double arrow open parentheses b comma a close parentheses element of R H e n c e comma space open parentheses a comma b close parentheses element of R rightwards double arrow open parentheses b comma a close parentheses element of R. H e n c e comma space t h e space g i v e n space r e l a t i o n space i s space s y m m e t r i c.$

$T r a n s i t i v e colon I f space open parentheses a comma b close parentheses element of R space a n d space space open parentheses b comma c close parentheses element of R rightwards double arrow a plus b equals 2 P space a n d space b plus c equals 2 Q A d d i n g space b o t h space comma space w e space g e t a plus c plus 2 b equals 2 open parentheses P plus Q close parentheses rightwards double arrow a plus c equals 2 open parentheses P plus Q close parentheses minus 2 b rightwards double arrow a plus c space i s space a n space e v e n space n u m b e r. H e n c e comma space open parentheses a comma b close parentheses element of R space a n d space space open parentheses b comma c close parentheses element of R space rightwards double arrow open parentheses a comma c close parentheses element of R H e n c e comma space t h e space g i v e n space r e l a t i o n space i s space t r a n s i t i v e S i n c e space t h e space g i v e n space r e l a t i o n space i s space r e f l e x i v e comma space s y m m e t r i c space a n d space t r a n s i t i v e comma space i t space i s space a n space e q u i v a l e n c e space r e l a t i o n.$

Answered by satyajit samal | 31st May, 2015, 02:48: PM

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