# Help with relations/partitions

• Feb 6th 2011, 10:40 AM
steph3824
Help with relations/partitions
Let A={1....10} and let K=AxA. Define the relation R on K by ((x,y), (z,t)) ∈ R <===> x+y=z+t.
R induces a partition on K. For each cell of this partition give a representative as well as the cardinality of the cell.

I really need some help on this. What does it mean by "each cell of the partition"? What is the partition and what exactly is a cell?
• Feb 6th 2011, 11:01 AM
emakarov
From Wikipedia:
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A partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X.
Thus, "parts" or "cells" are subsets of the set that is being partitioned, K in this case. If a partition is generated by an equivalence relation, as in this case, partition "parts" are also called equivalence classes.

To understand the relation R better, I suggest drawing a 10 x 10 grid on the coordinate plane and connecting points that are related by R. Note that if two points (x1,y1) and (x2,y2) are related, then they have the same sum of their coordinates. What does it say about their location?
• Feb 6th 2011, 11:42 AM
Plato
To add a bit more for finite set. All partitions of a set correspond to an equivalence relation. In fact, there is one-to-one correspondence between the partitions of a set and equivalence relations on the set.

In this case two pairs are related if and only their coordinates add to the same number.
\$\displaystyle \{(1,4),(4,1),(2,3),(3,2)\}\$ is one equivalence class.
We say it the equivalence class determined by any of its members.
To finish, you need to find all classes. You are done when all 100 pairs are accounted for.
• Feb 6th 2011, 03:02 PM
steph3824
Will each of my classes contain an even number of members since the relation is about a PAIR of coordinates?

Also, what is meant by representative?

I also am not sure I fully understand what a cell is..in Plato's example, is each coordinate listed considered to be a cell?..meaning that the cardinality of that equivalence class is 4? Let me know if this is correct or not
• Feb 7th 2011, 12:00 AM
emakarov
Quote:

Originally Posted by steph3824
Will each of my classes contain an even number of members since the relation is about a PAIR of coordinates?

No, I don't see the connection. Indiviual coordinates are not members of equivalence classes, pairs of coordinates are. For example, {(3,1), (2,2), (1,3)} contains three members.

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Also, what is meant by representative?
A representative is just a member, or an element.

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I also am not sure I fully understand what a cell is.
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Originally Posted by emakarov
Thus, "parts" or "cells" are subsets of the set that is being partitioned, K in this case.

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in Plato's example, is each coordinate listed considered to be a cell?
No, coordinates, which are numbers from 1 to 10, are parts of ordered pairs, such as (1,2), and pairs are elements of cells, or equivalence classes.
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meaning that the cardinality of that equivalence class is 4?
Yes, though that class, or cell, contains pairs of coordinates, not individual coordinates and certainly not cells.