Describe the partition for the equivalence relation T

In summary: For the first one, every number in a subset is only in one subset once. For the second one, the numbers 0, 1/2, 5/4, and pi are equivalent to each other, but not to 3/2 or 7/4.
  • #1
needhelp83
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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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  • #2


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?
 
  • #3


(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]
 
  • #4


needhelp83 said:
(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]?
 
  • #5


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
 

1. What is the definition of an equivalence relation?

An equivalence relation is a relation between two elements that is reflexive, symmetric, and transitive. This means that for any element a, it is related to itself (reflexive), if a is related to b then b is related to a (symmetric), and if a is related to b and b is related to c, then a is also related to c (transitive).

2. How is a partition related to an equivalence relation?

A partition is a way of dividing a set into non-overlapping subsets. In the context of an equivalence relation, the elements in each subset are related to each other, but not to elements in other subsets. This means that the subsets created by a partition correspond to the equivalence classes of the equivalence relation.

3. Can you give an example of a partition for an equivalence relation?

One example of a partition for an equivalence relation is the set of integers divided into even and odd numbers. Each integer is either even or odd, and any integer is related to itself (reflexive) and to other integers of the same parity (symmetric and transitive).

4. How can you determine if a partition is valid for an equivalence relation?

To determine if a partition is valid for an equivalence relation, you can check if the subsets created by the partition are non-empty, disjoint, and cover the entire set. Additionally, the relation between elements in each subset should satisfy the properties of reflexivity, symmetry, and transitivity.

5. Why is understanding the partition for an equivalence relation important?

Understanding the partition for an equivalence relation is important because it allows us to group elements together based on their relation to each other, rather than their individual properties. This can be useful in various mathematical and scientific contexts, such as in graph theory, computer science, and statistics.

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