EQUIVALENCE RELATIONS
Recall that "a equiv_n b" means n  ba, i.e., there exists an integer k such that ba = nk.
One can think of this in terms of "sticks". Consider a stick of length n (cms, say). If we can lay out that
stick over and over, starting at a and ending exactly at b, then a and b are a multiple of n apart. All the
integers that can be reached this way, starting from integer a, are those that are "equivalent to a mod n".
This concept  as we saw much earlier  partitions Z into the subsets
nZ
1+nZ
2+nZ
...
(n1) + nZ
When n is 3, we get
3Z: {.
.. 6 3 0 3 6 .
..}  multiples of 3, aka integers equiv to 0 mod 3
1+3Z: {.
.. 5 2 1 4 7 .
..}  one more than multiples of 3, aka integers equiv to 1 mod 3
2+3Z: {.
.. 4 1 2 5 8 .
..}  two more than multiples of 3, aka integers equiv to 2 mod 3
This partition of Z is { 3Z, 1+3Z, 2+3Z }. Every integer is in one  and only one  of the three partition
elements. The integers within any *one* of those subsets are considered "equivalent" to one another: they
are all reachable from one another by sticks of length n; put differently, they all give the same remainder
when divided by n. Elements of a single partition subset are "the same" in some respect.
Such a situation can vastly be generalized to *any* partition of *any* set.
Definition
: Let S be a set, and P = (S1, S2, .
.. } a partition of S. We can consider the elements of any one
partition subset as
equivalent
to each other, in this sense: they all belong to the same partition subset!
Imagine S as a large set of people, and P as some breaking up of S into "teams"; this can be done in any
way at all, even at random. But once the teams are formed, members of the same team are the same in that
sense: they belong to the same team.
Here is another illustration: Let S = {0,1,2,.
..,9,10} and let P consist of the subsets
{0,9}
{2,4,6,7}
{1,8}
{3}
{5,10}
Then P partitions S: the subsets are disjoint and their union is S. And this means that  as far as this
partition goes  2, 4,6, and 7 are all equivalent to each other; also 5 and 10; 1 and 8; and 0 and 9. And 3 is
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 Summer '08
 staff
 Equivalence relation, partition subset

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