What is an Equivalence Relation? | Reflexive, Symmetric, and Transitive Properties
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- เผยแพร่เมื่อ 2 ก.พ. 2019
- What are equivalence relations? Equivalence relations are very nice relations to work with, and we are very familiar with a particular equivalence relation: equals! It is at times important to be able to recognize other equivalence relations, so we discuss what they are in today's math video lesson!
An equivalence relation is a binary relation that has three particular properties. A binary relation is basically a relation that relates two objects at a time, most of the relations we are familiar with are binary relations: like equal to and less than.
The three properties a binary relation has to have to be an equivalence relation are the reflexive, symmetric, and transitive properties.
We will call our relation R. The relation R is reflexive if every object under consideration relates to itself. The relation R is symmetric if it is true, for all objects under consideration, that when a relates to b, b also relates to a. The relation R is transitive if it is true, for all objects under consideration, that when a relates to b, and b relates to c, it is also true that a relates to c.
As an example of the reflexive property, 4 = 4. As an example of the symmetric property, 4 = 2*2 and 2*2 = 4. As an example of the transitive property, 8 = 4*2 and 4*2 = 2*2*2, and it is true that 8 = 2*2*2.
A relation that does not have some of these properties is less than on the real numbers. Notice that 3 is not less than 3, so it is not a reflexive relation. Also, 3 is less than 4, but 4 is not less than 3, so it is not a symmetric relation.
An example of a relation that breaks transitivity is the relation "is the father of". If Starbuck is the father of Stubb, and Stubb is the father of Flask, it is not true that Starbuck is the father of Flask, thus the relation is not transitive.
For a more mathematical example, consider the relation "set membership" or "is an element of". The number 2 is an element of the set { 1, 2 }. The set { 1, 2 } is an element of the set { { 1, 2 }, 4 }. But 2 is not an element of the set { {1, 2}, 4 }
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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Thanks. Computer programmer here trying to read a research paper and struggling. This video was really clear.
This vids is too underrated
It's a gem
Thank you! If you know anyone else who would find it helpful, sharing is a big help!
Thank you sir! Wish you were my professor
Amazing your explanation. Thank you for it!
Glad to help - thanks for watching!
Many thanks, I really appreciate your help. Best wishes to you. It was more pleasance of math, rather than wrath of math :)
That's good to hear, haha! Thanks for watching!
Amazing!
If 'aRb' and 'bRa'.. does the conclusion 'aRa' show transitivity? Or does transitivity only apply to a minimum of 3 objects?
Excellent
what a great video
Can you do a video likely exams questions on reflexive,symmetric and transitive questions answer then?
Give me questions on this
Do a comprehensive class on this
Can you please do a video on equivalence classes, it’s so confusing to me how [x] = the set of all elements related by x to y but solving for the set makes no sense to me.
Hi brother, may I pose a question: let’s say we have an equivalence relation aRb. Why can’t I represent this within set theory as set T comprising subset of Cartesian product of a and b, mapped to a set U which contains true or false? Thanks so much!!
How we find x=2y+1 is reflexive symmetry and transitive
So let me get this straight. The goal behind equivalence relation is to introduce the concept of ‘=‘? As in what it means?
Im sorry if this is a dumb question but can someone explain the application for this?
Why does it have to meet all 3 properties (reflexive, symmetric, trasitive)?
Thank you very much bro 👍
Welcome 👍
Love your presentations and may I ask a couple humble questions friend:
1)
Can I say a “subset relation” is an equivalence relation since a set is its own subset so it satisfies the reflexive property. But what about the symmetric ? Can I say aRa and trivially aRa so it’s symmetric too? As for transitive this is where I am confusion: if we have a b and c, can the a b and c all be the same object? Like they all are the object “a” so then a is a subset of a and a is a subset of a then a is a subset of a therefor transitive? But if it’s not all “a’s” then we cannot say for sure it’s gonna be transitive right?!
2)
I been perusing differing TH-cam videos and it seems there are two completely different equivalence relation definitions. Is this because one has to do with set theory and one has to just do with something more basic? Or maybe I’m confused and there is only one definition but it’s explained differently ?
Edit: I should have put “{}” around the “a’s” since I was tryna denote sets.
Brother it is really great video i think u should make videos on the whole function chapter anf calculus
Thanks a lot, Tarun, and I have a lot more lessons on functions, as a sort of first chapter of calculus, on the way! Let me know if you ever have any video requests!
@@WrathofMath no i start watching your great videos from nowdays
But seriously u are great i think you are so passionate in maths u dont need improve ment but u should be more informetive i mean more point to point topics and learning
So the only mathematical comparator ( I don't know if that's the correct way to say it, but the =, >,
Thanks for watching and good question! Late reply, but better late than never! There are countless mathematical relations. Among the basic ones that you listed, you're right that != is the only one that is not transitive. But there are plenty others outside of those that are not transitive.
For example, the successor relation is not transitive. We can say 3 is the successor of 2, because 3 = 2+1, or 3 = S(2) where S is the successor function. Also, 2 is the successor of 1, since 2 = 1 + 1, or 2 = S(1). However, 3 is not the successor of 1.
Kind sir: I have one other question: I do hope you get to both my questions; why is the different than not a transitive? I can say 6 is different from 4 , 4 is different from 3, and 6 is different from 3.
ty
You're welcome, thanks for watching!
hi, im new elementary topology, trying to understand homeomorphisms, path components, etc. do you have videos re these topics? thank you
I have videos on group homomorphisms, but I do not have anything on topology specifically. Once the channel has grown enough - I'll be able to spend time making videos on everything!
Must R be a relation on set A only
Can't be a relation between two sets????
Thanks for watching and a relation is commonly thought to relate elements of a set A to some other set B, and we can equivalently think of this sort of relation as being a subset of the set AxB. You may see a relation being defined as a subset of AxA for some set A. If we have elements in some set B which are also involved in the relation, we could say A U B = C and the relation is a subset of CxC. The point is our single set can include whatever elements we want. So, at times it is convenient, but it is never necessary to describe a relation as being a subset of the cartesian product of two DIFFERENT sets. We can always consider it as a subset of the cartesian product of a single set. Does that help? The short answer to your question is yes.
@@WrathofMath yes I understand that but my question was that (is the equivalence relations is on the cartesian product of a single set only ??)
Is anti-symmentric relation an equavalance relation?
Thanks for watching and it depends on the set that the anti-symmetric relation is on. Many relations that aren't generally equivalence relations technically are on the right, usually small and trivial, set. Anti-symmetric relations typically will not be equivalence relations because, as the name suggests: as long two distinct elements relate to each other in one direction (like x relates to y) under an anti-symmetric relation, they cannot relate in the other direction (y cannot relate to x), in this way anti-symmetric relations cannot be symmetric and are thus not equivalence relations.
But remember it's not ALWAYS true. For example, consider the relation { (0,0) }. This means 0 relates to 0 under this relation, and that is it. This relation is technically antisymmetric because under this relation there are not distinct objects x and y that relate symmetrically. It is also an equivalence relation which you could easily verify. Does that make sense?
@@WrathofMath yes yes. It does
Would you kindly explain how (0,0) is an equivalence relation? For transitivity don’t we need an a b and c? We only have a and b so how do we even test transitivity?
Looking for a tutor?
I can do it
Got some relations basic videos man
I did have a lesson on relations, but then I took it down because I didn’t think it was good. Then I made another one, but apparently I took that down too because I didn’t think it was good. Are you looking for a video introducing relations with set theory or without set theory? I’d be happy to make one for you either way, third time is the charm!
Yes I am amigo I require it for school
A little late, but I hope it is useful! th-cam.com/video/HXFHVRS1ZW8/w-d-xo.html