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Topic: Sets Forming Partitions
Replies: 6   Last Post: Oct 7, 2012 6:19 PM

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Ben Brink

Posts: 198
From: Rosenberg, TX
Registered: 11/11/06
RE: Sets Forming Partitions
Posted: Oct 6, 2012 6:25 PM
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Edward,
Since (a) gives a partition, notice that numbers in the same class are equivalent. Therefore the ordered pairs in the relation must incude (1,1), (3,3), (1,3) and (3,1). Which pairs come from {5,4}? From {2,6}?
In (b), notice that 3 is in the first class and also in the second. What part of the definition of "partition" does that violate?
In (c), from the third equivalence class our pairs must include (4,4), (3,3), (6,6), (4,3), (3,4), (4,6), (6,4), (3,6) and (6,3). Do you see why? But from the first class, what's the only pair involving 1? What's the only pair we get from the second class? From the third?
In (d), are you sure of your answer? Hint: Which class has 5 in it?
Thanks for a great problem!
Ben

> Date: Sat, 6 Oct 2012 17:38:12 -0400
> From: discussions@mathforum.org
> To: discretemath@mathforum.org
> Subject: Sets Forming Partitions
>
> So I have been learning about binary and equivalence relations and things have been going smoothly. I have a question about partitions though because I'm not sure how to finish this problem:
>
> 2. Which of the following collections of subsets of S={1,2,3,4,5,6} form a partition of S? If the given collection of sets does form a partition, then list the ordered pairs if the equivalence relation produced by the partition. If the given collection does not form a partition, then explain which part(s) of the definition of partition fails.
> (a.) {1,3}, {5,4}, {2,6}
> This is a partition of S.
> (b.) {1,2,3}, {3,4}, {5,6}
> This is not a partition of S.
> (c.) {1}, {2}, {5}, {4,3,6}
> This is a partition of S.
> (d.) {1,3,4}, {2,6}
> This is a partition of S.
>
> Under each letter I figured out if the sets form a partition of S but I don't understand how to then list the ordered pairs if the equivalence relation produced by the partition if it is or explain which part(s) of the definition of partition fails. I'm halfway there I just need to understand how I to prove my answer..... If somebody could help me understand this I would really appreciate it





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