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Math Help - please help me with equivalence relations

  1. #1
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    please help me with equivalence relations

    We define a relation R on Z by setting xRy if and only if x-y divisible by 3.

    Find all the equivalence classess of R

    someone can help me to solve me this? and explain me a bit with words what r u doing....

    thnx
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  2. #2
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    Quote Originally Posted by tukilala View Post
    We define a relation R on Z by setting xRy if and only if x-y divisible by 3. Find all the equivalence classess of R.
    There are three equivalence classess:
    \begin{gathered}<br />
\left\{ { \cdots , - 9, - 6, - 3,0,3,6,9, \cdots } \right\} \hfill \\<br />
\left\{ { \cdots , - 8, - 5, - 2,1,4,7, \cdots } \right\} \hfill \\<br />
\left\{ { \cdots , - 7, - 4, - 1,2,5,8, \cdots } \right\} \hfill \\ <br />
\end{gathered} .

    Can you explain why?
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  3. #3
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    but why????

    what is common to
    ......1,4,7..........
    ......2,5,8..........
    ......3,6,9.......

    ????
    why this is the three equivalence classess????
    what is special about each class?????
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  4. #4
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    Equivalence Classes

    Hello tukilala -

    Quote Originally Posted by tukilala View Post
    We define a relation R on Z by setting xRy if and only if x-y divisible by 3.

    Find all the equivalence classess of R

    someone can help me to solve me this? and explain me a bit with words what r u doing....

    thnx
    When an equivalence relation R is defined on a set A, it partitions set A into equivalence classes.

    Let's break that sentence down a bit. What does R partitions A mean? Well, it means that R divides the whole of set A into disjoint (non-overlapping) subsets. These subsets are called equivalence classes.

    Why are they called equivalence classes? Because any one of these subsets contains only elements that are equivalent to one another. In other words, x and y are two elements from the same equivalence class if and only if xRy.

    As an example, suppose set A
    is {human beings} and R is the relation "has the same birthday (month and day) as". Then it's fairly obvious that R is reflexive (you have the same birthday as yourself), symmetric (if x has the same birthday as y, then y has the same birthday as x) and transitive (if x has the same birthday as y, and y has the same birthday as z, then x has the same birthday as z). So R is an equivalence relation. What are its equivalence classes? Well, of course, there's an equivalence class for each day of the year - 365 of them altogether (or even 366, if you count February 29!). Each class contains all the people whose birthdays are on that particular day.

    Now, what about your question? R is the equivalence relation on \mathbb{Z} which is such that xRy if and only if (x-y) is divisible by 3. What this means is that xRy if and only if x and y leave the same remainder when divided by 3.

    Can you see why? Suppose x leaves a remainder r_1 and y a remainder r_2 when they are divided by 3. Then, for some m, n \in \mathbb{Z}:

    x=3m+r_1 and y=3n+r_2 (This means that m and n are the quotients when x and y are divided by 3.)

    So x-y=(3m+r_1)-(3n+r_2)

    =3(m-n)+(r_1-r_2)

    Two things then follow:

    (1) If x-y is divisible by 3, then r_1-r_2=0, or r_1=r_2
    . In other words, x and y must leave the same remainder when divided by 3.

    (2) If r_1=r_2, then x-y=3(m-n). In which case, x-y is divisible by 3.

    In other words, x-y is divisible by 3 if and only if x and y leave the same remainder when divided by 3.

    OK so far?

    Now we can think about the equivalence classes of
    R. The most important question is: what are the possible remainders when a number is divided by 3? The answers, of course, are 0, 1 and 2. This means that there will be:

    • one equivalence class for all the numbers that leave a remainder 0;
    • one equivalence class for all the numbers that leave a remainder 1;
    • and one equivalence class for all the numbers that leave a remainder 2.

    There aren't any other remainders, so every whole number must appear in one of these three classes.

    Can you see now where Plato's answers came from?

    Grandad
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