Show that the number of equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) is two.

Asked by abhinavsaini123 | 1st Jun, 2015, 08:50: PM

Expert Answer:

T h e space s m a l l e s t space e q u i v a l e n c e space r e l a t i o n space o n space t h e space s e t space open curly brackets 1 comma 2 comma 3 close curly brackets space c o n t a i n i n g space left parenthesis 1 comma 2 right parenthesis space a n d space left parenthesis 2 comma 1 right parenthesis space i s space S equals open curly brackets left parenthesis 1 comma 1 right parenthesis comma space left parenthesis 2 comma 2 right parenthesis space comma space left parenthesis 3 comma 3 right parenthesis comma space left parenthesis 1 comma 2 right parenthesis comma space left parenthesis 2 comma 1 right parenthesis close curly brackets N o w space t h e r e space a r e space 4 space p a i r s space l e f t space n a m e l y space left parenthesis 1 comma 3 right parenthesis comma space left parenthesis 3 comma 1 right parenthesis comma space left parenthesis 2 comma 3 right parenthesis comma space left parenthesis 3 comma 2 right parenthesis  L e t space u s space s u p p o s e space w e space a d d space left parenthesis 1 comma 3 right parenthesis space t o space S. T h e r e f o r e comma space w e space h a v e space t o space a d d space left parenthesis 3 comma 1 right parenthesis space t o space p r o v e space s y m m e t r y. N o w space t o space p r o v e space t r a n s i t i v i t y comma space w e space h a v e space t o space a d d space left parenthesis 2 comma 3 right parenthesis space a n d space left parenthesis 3 comma 2 right parenthesis space a l s o. H e n c e comma space t h e space e q u i v a l e n c e space r e l a t i o n space g r e a t e r space t h a n space S space i s space t h e space u n i v e r s a l space r e l a t i o n.  therefore t h e space n u m b e r space o f space e q u i v a l e n c e space r e l a t i o n space i s space 2.

Answered by Prasenjit Paul | 2nd Jun, 2015, 10:41: AM