ChapterZero

Equivalence relations

Filed under: Mathematics — Alex @ 9:51 pm 3/27/2005

I’m having a problem with an algebra exercise, of all things:

Define, by means of a partition, an e.r. on $\Z$ which has, for each positive integer , exactly one equivalence class with elements. Describe this equivalence relation in the form ‘ iff $\ldots$ ’

Update:
Got it! But there has got to be an easier one…

$p_1 = \{0\}$
$p_2 = \{-1, -2\}$
$p_{2n} = \{-(n^2 + n), \ldots, -n^2 + n -1 \}$
$p_{2n+1} = \{n^2, n^2 + 2n\}$

Then $P = \{p_m\}_{m \in \N}$ is a partition of $\Z$ . Visually, you construct it by defining the 1 element set to contain only 0, then the sets containing odd elements cover the positive integers in increasing order, and the sets containing even elements cover the negative integers. So:
$p_2 = \{-1, -2\} \quad p_3 = \{1, 2, 3\} \quad p_4 = \{-3, -4, -5, -6\} \quad p_5 = \{4, 5, 6,7,8\}$

Is there a simpler one? Since I did this assignment a whole two days before it’s due, I have time to pick some smarter people’s brains tomorrow.

Update again;

Turns out I misread the homework assignment: that problem wasn’t assigned after all.

Equivalence relations

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