Equivalence relations

March 27th, 2005 ~ Posted in: Mathematics

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 n, exactly one equivalence class with n elements. Describe this equivalence relation in the form ‘ x R y 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.

Mathematics ~

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