#equivalence class

What is relation? Explain equivalence relation with the help of an example?

Relation-Sometimes we need to establish relation s between two or more sets. For example, a software development company has a set of specialists in different technology domains, or a company gets some projects to develop. Here the company needs to establish a relation between professionals and the project in which they will participate. To solve this type of problem the following concepts are…

In one of the previous posts where I had explained how to construct rational numbers, I had explained the concept of division in the perspective of equivalence class and by using the concept we had redefined how to understand the concept of rational numbers.

Now we are going to understand and give a better look at what addition and multiplication of rational numbers stand for.

Now let’s ask ourselves a very basic question.

What exactly is addition of two rational numbers ?

And even when you are really familiar with the concept of addition I beg you to think about it in a different way and we can do this by considering a few examples -

Let’s consider , (a/b) + (c/d) = (a+c)/(b+d).

I know this looks really wrong or different than what you had learnt about addition but for the sake of example let’s try to understand this process of addition and at the end we will know what is meant by a well defined operation .Let’s crunch values for a,b,c & d and let’s look at what this operation results in.

For a=1 ,b=2 ,c=4 ,d=5 | (a/b)+(c/d)=(a+c/b+d) == (1/2)+(4/5)=(5/7).

Now let’s consider values of a & b which are in the same equivalent class as that of (1,2) which represents ½.

For a=2 ,b=4 ,c=4 ,d=5 | (a/b)+(c/d)=(a+c/b+d) == (2/4)+(4/5)=(6/9).

Now this is odd , because even when the given input to the operation for the equivalence class is the same , even then the output doesn’t belongs to the same equivalence class as (6,9) (which actually belongs to the equivalence class of (2,3)) doesn’t belongs to the same equivalent class as that of (5,7) (which actually belongs to the equivalence class of (5,7)) and this is what is being called as “not well defined operation” and why so ?? Because the output equivalence class is dependent on what input is provided.

For an operation to be well defined , the output should be in the same equivalent class even when the value of input changes amongst the same equivalent class.

Let’s consider one more example -

(a/b) + (c/d) = 3/2.

Now in this case , the operation is once again the addition we are familiar with , but this operation is well defined as the equivalence class of the output is not dependent upon the values of input. Heck , it’s independent of any sort ot input provided.

And finally let’s look at the final example which is the way of addition , most of the people do addition of two rational numbers.

(a/b)+(c/d)=(ad+bc)/(bd).

But , wait, Is it a well defined operation ??

We can check it  by doing the following -

If (a,b) ~ (a’,b’) [When (a,b) is equivalent to (a’,b’) or a/b=a’/b’]

& (c,d) ~ (c’,d’) [When (c,d) is equivalent to (c’,d’) or c/d=c’/d’].

So if the definition of addition as shown above is equivalent then,

(ad+bc,bd) ~ (a’d’+b’c’,b’d’).

And here’s a really simple proof for showing it -

(a,b) ~ (a’,b’) could be represented by a/b=a’/b’.

(c,d) ~ (c’,d’) could be represented by c/d=c’/d’.

(ad+bc,bd) could be represented by (ad+bc)/(bd).

replacing b by ab’/a’ and d by cd’/c’ we get,

(ad+bc)/(bd).

=(acd’/c’ + cab’/a’)/(ab’/a’)(cd’/c’).

Now I am going to get extremely lazy here and leave the rest of steps for you to calculate , but the final step will result in -

=(a’d’+b’c’)/(b’d’) which could be represented by (a’d’+b’c’,b’d’).

Hence,

(ad+bc,bd)~(a’d’+b’c’,b’d’).

Now we are aware of the fact that the we can actually understand the concept of a binary operation like addition in the perspective of equivalence class. Now let’s try to construct and understand what multiplication of two rational numbers will be.

But, we can actually use the way how we defined addition in the previous portion and we can define multiplication by -

If (a,b) ~ (a’,b’) [When (a,b) is equivalent to (a’,b’) or a/b=a’/b’]

& (c,d) ~ (c’,d’) [When (c,d) is equivalent to (c’,d’) or c/d=c’/d’].

then (ac,bd)~(a’c’,b’d’) [Which basically represents (ac)/(bd)=(a’c’)/(b’d’)].

And once again I am going to be lazy enough to leave it’s proof for you.

What is a coset?

a shifted subgroup

like −5, −2, 1, 4, 7, 10

We want to take theories and turn them over and over in our hands, turn the pants inside out and look at the sewing; hold them upside down; see things from every angle; and sometimes, to quotient or equivalence-class over some property to either consider a subset of cases for which a conclusion can be drawn (e.g., "all fair economic transactions" (non-exploitive?) or "all supply-demand curveses such that how much you get paid is in proportion to how much you contributed" (how to define it? vary the S or the D and get a local proportionality of PS:TS? how to vary them?)

Consider abstractly a set like {a, b, c, d}. 4! ways to rearrange the letters. Since sets are unordered we could call it as well the quotient of all rearangements of quadruples of once-and-yes-used letters (b,d,c,a). /p>

Descartes' concept of a mapping is "to assign" (although it's not specified who is doing the assigning; just some categorical/universal ellipsis of agency) members of one set to members of another set.

For example the Hash Map of programming.

{ '_why' => 'famous programmer', 'North Dakota' => 'cold place', ... }

Or to round up ⌈num⌉: not injective because many decimals are written onto the same integer.

Or to "multiply by zero" i.e. "erase" or "throw everything away":

Old programmers don't die, they're piped to /dev/null.

— protea (@isomorphisms) January 2, 2013

Linear maps with det=0 :: - | /dev/null :: matter into a black hole :: (or, on Earth, a trash grinder)

— protea (@isomorphisms) January 2, 2013

In this sense a bijection from the same domain to itself is simply a different--but equivalent--way of looking at the same thing. I could rename A=1,B=2,C=3,D=4 or rename A='Elsa',B='Baobab',C=√5,D=Hypathia and end with the same conclusion or "same structure". For example. But beyond renamings we are also interested in different ways of fitting the puzzle pieces together. The green triangle of the wooden block puzzle could fit in three rotations (or is it six rotations? or infinity right-or-left-rotations?) into the same hole.

By considering all such mappings, dividing them up, focussing on the easier classes; classifying the types at all; finding (or imposing) order|pattern on what seems too chaotic or hard to predict (viz, economics) more clarity or at least less stupidity might be found.

The hope isn't completely without support either: Quine explained what is a number with an equivalence class of sets; Tymoczko described the space of musical chords with a quotient of a manifold; PDE's (read: practical engineering application) solved or better geometrically understood with bijections; Gauss added 1+2+3+...+99+100 in two easy steps rather than ninety-nine with a bijection; ....

  It's hard for me to speak to why we want groups and what they are both at once. Today I felt more capable of writing what they are.

So this is the concept of sameness, let's discuss just linear planes (or, hyperplanes) and countable sets of individual things.

Leave it up to you or for me later, to enumerate the things from life or the physical world that "look like" these pure mathematical things, and are therefore amenable by metaphor and application of proved results, to the group theory.

But just as one motivating example: it doesn't matter whether I call my coordinates in the mechanical world of physics (x,y,z) or (y,x,z). This is just a renaming or bijection from {1,2,3} onto itself.

Even more, I could orient the axis any way that I want. As long as the three are mutually perpendicular each to the other, the origin can be anywhere (invariance under an affine mapping -- we can equivalence-class those together) and the rotation of the 3-D system can be anything. Stand in front of the class as the teacher, upside down, oriented so that one of the dimensions helpfully disappears as you fly straight forward (or two dimensions disappear as you run straight forward on a flat road). Which is an observation taken for granted by my 8th grade physics teacher. But in the language of group theory means we can equivalence-class over the special linear group of 3-by-3 matrices that leave volume the same. Any rotation in 3-D

Sameness-preserving Groups partition into:

permutation groups, or rearrangements of countable things, and

linear groups, or "trivial" "unimportant" "invariant" changes to continua (such as rescaling--if we added a "0" to the end of all your currency nothing would change)

conjunctions of smaller groups

The linear groups--get ready for it--can all be represented as matrices! This is why matrices are considered mathematically "important". Because we have already conceived this huge logical primitive that (in part) explains the Universe (groups) -- or at least allows us to quotient away large classes of phenomena -- and it's reducible to something that's completely understood! Namely, matrices with entries coming from corpora (fields).

So if you can classify (bonus if human beings can understand the classification in intuitive ways) all the qualitatively different types of Matrices,

then you not only know where your engineering numerical computation is going, but you have understood something fundamental about the logical primitives of the Universe!

Aaaaaand, matrices can be computed on this fantastic invention called a computer!

homotopy

When I was a maths teacher some curious students (Fez and Andrew) asked, "Does i, √−1, exist? Does infinity ∞ exist?" I told this story.

You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession -- Peano map, Peano map, Peano map, Peano map. Sure. But that's an example of the number 4, not the number 4 itself.

So is it even possible to say what a number is? No, let's ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.

Willard van Orman Quine had an interesting answer. He said that the number seventeen "is" the equivalence class of all sets of with 17 elements.

Accept that or not, it's at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and ∞ serve a practical purpose just like the concept of ⅓ (you know, the obvious moral cap on income tax). For instance

if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)

∞ allows for limits and therefore derivatives and calculus. Just one example apiece.