SETS RELATIONS
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Suppose A and B are sets, f : A -> B is a function, and R is the relation
on A so that for x, y in A, xRy if and only if f (x) = f (y).
1. Show that R is an equivalence relation on A.

2. Show that for every set A and every equivalence relation R on A, there is
a set B and a function f : A -> B such that R is the relation described above

GIVEN …F[A]=B
AND …X R Y IFF F[X]=F[Y]
1. REFLEXIVE ..
WE HAVE F[X]=F[X]
SO X R X …OK
2. SYMMETRIC ..
X R Y …SO …F[X]=F[Y]=F[X] …
SO Y R X …OK
3.TRANSITIVE ..
X R Y & Y R Z …SO …
F[X]=F[Y] & F[Y] = F[Z]…THAT IS F[X]=F[Z]
SO X R Z ……OK
1 , 2 AND 3 IMPLY THAT R IS AN EQUIVALENCE RELATION …
GIVEN A SET A & A RELATION R AS ABOVE IN A .
TST ...THERE IS SET B & A FUNCTION F[A]=B AS ABOVE .
1. R IS REFLEXIVE
SO IF X IS ANY ELEMENT IN A THEN F[X]=F[X]
2. R IS SYMMETRIC …
SO IF X R Y , THEN IT IMPLIES Y R X …
THAT IS ..
F[X]=F[Y]=F[X]
3.R IS TRANSITIVE …
SO IF X R Y AND Y R Z , THEN IT IMPLIES …X R Z ….THAT IS ..
F[X] = F[Y] = F[Z]
HENCE IF WE DEFINE A SET SAY B = F[A] , THEN FROM 1,2,3 ABOVE
THEY IMPLY THAT F[X] IS UNIQUE , ON TO & ONE TO ONE ..
THAT IS THE EQUIVALENCE RELATION REPRESENTS A BIJECTIVE FUNCTION
HENCE THE RESULT ..