# RELS - EQUIVALENCE RELATIONS Recall that"a equiv_n b means...

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EQUIVALENCE RELATIONS Recall that "a equiv_n b" means n | b-a, i.e., there exists an integer k such that b-a = 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 ... (n-1) + 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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RELS - EQUIVALENCE RELATIONS Recall that"a equiv_n b means...

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