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Describe the partition for the equivalence relation T

by needhelp83
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needhelp83
#1
Jun30-08, 02:10 AM
P: 199
For the set A = {1,2,3,4,5,6,7}, determine whether script A is a partition of A. script A = {{1,3,},{5,6}, {2,4},{7}}

Describe the partition for the equivalence relation T defined for x,y [tex] \in \mathbbc{R} [/tex] by X T y iff [tex] \left[ \left[x \right] \right] = \left[ \left[y \right] \right] where \left[ \left[x \right] \right] [/tex] is definied to be the greatest integer iin x (the largest integer n such that n [tex] \leq [/tex] x).

Can anyone help me with this partition stuff. It would be very appreciated. :)
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CompuChip
#2
Jun30-08, 02:59 AM
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Can you start by giving the definition of a partition?
Then try to check if the given sets [itex]A, \mathcal{A}[/itex] satisfy this definition.

For the second one, can you imagine what the equivalence classes look like?
needhelp83
#3
Jun30-08, 06:47 AM
P: 199
(i) If X [tex] \in \mathcal{A}, [/tex] then X [tex] \neq \o [/tex]
(ii) If X [tex] \in \mathcal{A} [/tex] and Y [tex]\in \mathcal{A}[/tex], then X=Y or X [tex]\cap [/tex] Y= [tex]\o[/tex]
(iii)[tex]\bigcup_{X \in \mathal{A}}X=A[/tex]

I found this to be a definition for an equivalence class:
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:

[tex]\left[a\right]={x \in X|x \sim a} [/tex]

HallsofIvy
#4
Jun30-08, 08:10 AM
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Describe the partition for the equivalence relation T

Quote Quote by needhelp83 View Post
(i) If X [tex] \in \mathcal{A}, [/tex] then X [tex] \neq \o [/tex]
(ii) If X [tex] \in \mathcal{A} [/tex] and Y [tex]\in \mathcal{A}[/tex], then X=Y or X [tex]\cap [/tex] Y= [tex]\o[/tex]
(iii)[tex]\bigcup_{X \in \mathal{A}}X=A[/tex]
That's long winded! A "partition" of a set, A, is a collection of subsets of A such that every member of A is in one and only one of the subsets. Here, the members of A are 1, 2, 3, 4, 5, 6, 7. Is every one of those numbers in one of the given subsets? Is any number in more than one?

I found this to be a definition for an equivalence class:
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:

[tex]\left[a\right]={x \in X|x \sim a} [/tex]
Okay, and the relation x T y is defined by "x T y if and only if the largest integer less than or equal to x is the same as the largest integer less than or equal to y".

Now try some examples. What numbers are equivalent to 0? to 1/2? to 5/4? to [itex]\pi[/itex]?
needhelp83
#5
Jun30-08, 08:32 AM
P: 199
All the numbers in a subset are only in one subset once.

-What do you mean by what numbers are equivalent to 0, 1/2, pi, etc


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