Equivalence relation problem

- jango_IN_DTOWN

Equivalence relation problem

- jamiebookeater

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- jango_IN_DTOWN

##### 1 Attachment

- jango_IN_DTOWN

- ganeshie8

|dw:1444584458933:dw|

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## More answers

- ganeshie8

for each relation, we need to check :
1) reflexive
2) symmetry
3) transitive

- ganeshie8

for i :
reflexive :
\((x_1, y_1)\sim (x_1,y_1)\) because \(y_1=y_1\)
symmetry :
\( (x_1,y_1)\sim(x_2,y_2) \implies (x_2,y_2)\sim (x_1,y_1)\) because \(y_1=y_2 \implies y_2=y_1\)
transitivity :
\( (x_1,y_1)\sim(x_2,y_2) \) and \((x_2,y_2)\sim (x_3,y_3)\) \(\implies (x_1,y_1)\sim (x_3,y_3)\) because \(y_1=y_2\) and \(y_2=y_3\)\(\implies y_1=y_3\)
therefore, this is an equivalence relation

- jango_IN_DTOWN

the reflexive part in i) I am having a confusion

- ganeshie8

to better understand reflexivity,
maybe consider a relation that is not reflexive

- ganeshie8

first of all, as the name says, "reflexive" refers to the reflection that you see when u look at mirror. you see your own reflection...
we say a relation is reflexive if \(x\sim x\) for all \(x\) in the relation.

- ganeshie8

can you think of a relation that is not reflexive ?

- jango_IN_DTOWN

yes a>b

- ganeshie8

nice, another one : a-b is odd

- jango_IN_DTOWN

yeah

- ganeshie8

so do you get why the relation in part i is reflexive ?

- jango_IN_DTOWN

yes now it is clear..

- ganeshie8

review quick top 3 properties
|dw:1444585876121:dw|

- jango_IN_DTOWN

checked..

- ganeshie8

can you guess what the equivalence classes will be

- jango_IN_DTOWN

x will be any member of R and y will be fixed

- ganeshie8

Exactly! for example below is an equivalence class :
[(x, 1)] = {(1,1), (2,1), (2.2, 1), (-99, 1), ... }

- ganeshie8

below is another equivalence class
[(x, 3)] = {(1,3), (2,3), (2.2, 3), (-99, 3), ... }

- jango_IN_DTOWN

[(x,y)]={(a,b)belongs to R^2 such that b=y}

- ganeshie8

looks nice

- jango_IN_DTOWN

so this is the answer right?

- ganeshie8

Yes thats the answer for part i

- ganeshie8

look at the relation in part ii,
whats ur first guess, can it be an equivalence relation ?

- jango_IN_DTOWN

I think it is reflexive and symmetric but transitivity cant say

- ganeshie8

right, just show an example that its not transitive

- jango_IN_DTOWN

then we need to consider general points of R^2

- ganeshie8

yes just pick any one simple example

- jango_IN_DTOWN

(1,2)~(1,3) and(1,3)~(2,3) but (1,2) is not ~ to (2,3)

- ganeshie8

that will do, that proves the relation is not transitive
consequently its not equivalence relation

- jango_IN_DTOWN

yes correct and hence no question of equivalence classes arise

- ganeshie8

good
iii looks innocent, but it can be very tricky...

- ganeshie8

because there are several ways to get an integer by taking difference of two numbers :
3 - 1 = integer
1.4 - 0.4 = integer
0.3 - 0.3 = integer
..

- jango_IN_DTOWN

ohhhh

- ganeshie8

proving that it is an equivalence relation is trivial
but figuring out equivalence classes can be tricky...

- jango_IN_DTOWN

we see that the y component can be anything

- ganeshie8

hey wait, does it really pass transitivity ?

- jango_IN_DTOWN

yeah if a-b is an integer and b-c is an integer then a-c must be an integer

- jango_IN_DTOWN

a-c=(a-b)+(b-c)= sum of integers

- ganeshie8

Ahh thats clever! okay so it does pass transitivity

- jango_IN_DTOWN

yes.. and the symmetric and reflexive parts are also satisfied.. so we need to figure out the equivalence realtion

- ganeshie8

you mean equivalence `classes`

- jango_IN_DTOWN

yeah oops

- jango_IN_DTOWN

so our y component can be anything

- jango_IN_DTOWN

but x component will be those real numbers whose difference gives integer

- ganeshie8

How about this
[(x,y)]={(a,b)belongs to R^2 such that a = x-floor(x)+k, \(k \in \mathbb Z\)}

- ganeshie8

floor(x) gives the integer part of x
so, x - floor(x) gives the fractional part of x

- jango_IN_DTOWN

yeah it gives integer always I thought of another one, [(x,y)]={(a,b)belongs to R^2 such that (x-a) belongs to Z}

- ganeshie8

looks much better!

- jango_IN_DTOWN

But both will work

- ganeshie8

mine is like construction
your's is more like a logic statement
yeah both works, but yours looks better

- jango_IN_DTOWN

Thanks. so we are done with the questions. I just need to construct the language only..:)

- ganeshie8

np :)

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