Describe the partition for the equivalence relation T

[needhelp83]

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 $$\in \mathbbc{R}$$ by X T y iff $$\left[ \left[x \right] \right] = \left[ \left[y \right] \right] where \left[ \left[x \right] \right]$$ is definied to be the greatest integer iin x (the largest integer n such that n $$\leq$$ x).

Can anyone help me with this partition stuff. It would be very appreciated. :)

 

[CompuChip]

Can you start by giving the definition of a partition?
Then try to check if the given sets $A, \mathcal{A}$ satisfy this definition.

For the second one, can you imagine what the equivalence classes look like?

 

[needhelp83]

(i) If X $$\in \mathcal{A},$$ then X $$\neq \o$$
(ii) If X $$\in \mathcal{A}$$ and Y $$\in \mathcal{A}$$, then X=Y or X $$\cap$$ Y= $$\o$$
(iii)$$\bigcup_{X \in \mathal{A}}X=A$$

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:

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

 

[HallsofIvy]

(i) If X $$\in \mathcal{A},$$ then X $$\neq \o$$
(ii) If X $$\in \mathcal{A}$$ and Y $$\in \mathcal{A}$$, then X=Y or X $$\cap$$ Y= $$\o$$
(iii)$$\bigcup_{X \in \mathal{A}}X=A$$

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:

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

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 $\pi$?

 

[needhelp83]

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