Thursday January 22, 2009
Equivalence Relation is - a subset R - of A X A - if:
for all a,b,c belonging to A:
1. (a,a) belongs to R
2. (a,b) belongs to R => (b,a) belongs to R
3. (a,b) belongs to R and (b,c) belongs to R => (a,c) belongs to R
Equivalent notations:
1. a~a
2. a~b => b~a
3. a~b & b~c => a~c
1. aRa
2. aRb => bRa
3. aRb & bRa => aRc
Equivalence Relation (ER) is thus a relation, which relates 2 elements with some similarity.
With this following example, I hope to recall what Equivalence Relation is, in case I should forget:
Set A is set of all points on X-Y plane.
Thus A is: { (0,0),..
.
(1,0),..........(1,90),..........(1,180),...........
. . .
(2,0),............(2,90),.........(2,180),...............
. . .
. . .}
Let's define an equivalence relation which says that 2 points are related(equivalent) if they are equidistant from the origin.
Thus R is: { .................... ( (1,0),(1,90) ),.........( (1,90),(1,0) ),................
................( (1,90),(1,180) ).....................
.............................( (2,90),(2,0) )............................}
Note that it is not the relation (equivalency) amongst the members of R that were are concerned about. It is that each element of R is a pair of 2 related(equivalent) elements of A.
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