Discrete Math - 9.5.1 Equivalence Relations

Exploring a special kind of relation, called an equivalence relation. Equivalence classes and partitions are also discussed.
Video Chapters:
Introduction 0:00
Equivalence Relations 0:04
Is This an Equivalence Relation? Yes 4:14
Is This an Equivalence Relation? No 7:16
Equivalence Classes 10:51
Partitions 12:55
Partition of Integers mod 4 17:00
Find the Partitions 19:34
Up Next 22:13
Textbook: Rosen, Discrete Mathematics and Its Applications, 7e
Playlist: • Discrete Math I (Entir...

[@zzeynep2813]

I have watched from Turkey. Your examintaion is very helpful and clear. Thank you. I hope that I take high points from the midterm about Relations :)

[@luisdrg6995]

In example 1, Another way to find if it's Symmetric: aRb -> bRa (a - b) -> (b - a) Note that (b - a) can also be written as -(a - b). Since we assumed that (a - b) is an integer, we can say that -(a - b) is a negative integer. Therefore, it is symmetric :)

[@merlenye6347]

This is extremely helpful! Thank you!

[@ifeomaveronicaobialunamman2938]

I totally love your videos! Clearly understood!

[@polarbear986]

Woww, you are so generous for creating and sharing these videos🙏 Thank you so much. I'm having a discrete math exam soon. I find your teaching very helpful!!

[@brormaos8616]

thank you for the videos, teacher :)

[@johnywaves]

Thank you. thank you. thank you. You make this so easy to understand

[@ddharsh2548]

excellent explanation!!!

[@mathsplus01]

Very clear video. I can use this in my tutoring. Thank you

[@AhmedSalah-yd8di]

you are the best

[@nangalucia7lucia667]

Thanks very much. Ur video was helpful indeed

[@tarazjuszszulakiewicz2678]

12:51 - can we also write something like [a] = {a,a}, a is in Z since aside from pairs a=-b we also could have pairs a=b?

[@MathCuriousity]

Hey! If you have a moment, may I ask 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!!

[@berkouklahcene2594]

Professor i have a question, how to determine the number of equivalence classes in a set ?

[@rachelvanderlely6324]

You're the best!

[@SawFinMath]

Thanks!

[@NtsakoNkuna-fn9il]

Mam trans on the last part i dont get it isn't 2/16 a fraction a decimal Hence it is not an integer? Therfore the whole thing is not transative? 😭

[@omrifishbein7159]

In 16:45 a class [0] is referred to. Since 0 is not a member of the set this I think is a mistake. Should be for example [3] or [6]...

[@PoPo-tc8ir]

agree

[@panashemureyi2584]

so when you get to a[4] the values are the same with a[0]. 19:25

[@zeynepergul4768]

12 : 20 why didnt we think if a and b are equal to each otther and only write the oppossite values

[@gbemisolaagboola8255]

at 10:38 in Transitiv example , what rules says the number has to be 16?,what if its say 7

[@DevProTips]

It has to be some number that 8 divides (for example, 16, 24, 32, etc.). It is given that _a_ divides _b_ and _b_ divides _c._ Using that information, you need to be able to conclude that _a_ divides _c_ as well. [To show this you can say, _a_ goes into _b_ some number of times, _m,_ and _b_ goes into _c_ some number of times, _n,_ and because of that, the number of times _a_ goes into _c_ is _m*n._ Since two integers _m_ and _n_ multiplied together result in an integer, you can say _a_ divides _c_ and therefore the relation is transitive.

[@StoneColdMagic]

I loved your answer to my question about why. Because we are math dorks! :) However, I really would like to know if there is a real answer to this. I find it very common that fellow students say "I understand the concept, but what's the point?" And inevitably, the professor does not have an answer. I've asked professors at school. I've asked (comments) other KZbinrs, and nobody seems to have an answer. :( What's the real-world application for all this Discrete Math stuff? Thanks for all the great content!

[@makaylabullock2961]

Its actually very much similar to computer/programming logic. If you can understand the logic and functionality behind the programming/coding it's easier to put together your programs as well as take them apart and understand all the pieces that went into it. Currently, that is the direct application I've seen with it and used it for.

[@buh357]

You can apply discrete math to understand programming language.

[@StoneColdMagic]

@@buh357 - I get that. Hence my question. How do equivalence relations relate to / help us understand programming?

[@moski9861]

@@StoneColdMagic I hope you are satisfied with this answer from ChatGPT.: Equivalence relations can be useful in programming in various ways. Here are a few examples: Testing equality: In programming, we often need to compare two values to check if they are equal. Equivalence relations provide a formal definition of what it means for two objects to be equivalent, which helps us design and implement tests for equality. For example, in Python, the == operator tests for equality, which is based on the equivalence relation of identity (two objects are equal if they have the same identity) or value (two objects are equal if they have the same value). Grouping objects: In some cases, we want to group objects together based on their properties. Equivalence relations allow us to partition a set of objects into equivalence classes, where each class contains objects that are equivalent to each other. This is useful in data analysis, clustering, and other applications. For example, we might group a set of integers into equivalence classes based on their parity (even or odd). Object-oriented programming: In object-oriented programming, we often define classes that represent abstract concepts or entities. Equivalence relations can help us define the behavior of objects of a class. For example, we might define a class of geometric shapes and specify that two shapes are equivalent if they have the same area. Hashing: In some programming contexts, we need to store and retrieve objects quickly based on their values. Hashing is a technique that maps an object to a unique integer based on its value. Equivalence relations can help us design hashing functions that are consistent (two equivalent objects should have the same hash value) and efficient (the hash function should minimize collisions between different objects).

[@RA-hn1wm]

@@StoneColdMagici think it is like the arrays concept in java not really sure tho

[@StoneColdMagic]

Yet another excellent video. Thank you!. :) I find that it's a common theme when I study these topics . . . why? What's the point? What is the purpose of an equivalence relation, and what's the purpose of a partition? Why do we study them, and what are real world uses/examples of them? Thanks again!

[@SawFinMath]

Because we are math dorks?

[@PaulBoldyrev]

@@SawFinMath This earned a sub. Great video :)

[@StoneColdMagic]

@@SawFinMath - Funny. So true. However, this is a question I hear fellow students ask all the time, and I've never heard a good answer from another professor, so I'm curious what is the application for this stuff?

[@luisdrg6995]

Relations can be used to solve problems such as determining which pairs of cities are linked by airline flights in a network, finding a viable order for the different phases of a complicated project, or producing a useful way to store information in computer databases. In some computer languages, only the first 31 characters of the name of a variable matter. The relation consisting of ordered pairs of strings where the first string has the same initial 31 characters as the second string is an example of a special type of relation, known as an equivalence relation. Equivalence relations arise throughout mathematics and computer science. Copy & Paste straight from the textbook^^^^

[@bestyoueverhad.2408]

can you please explain the remainder 1 for 3 div 1, and remainder 2 for 3 div 2, im a bit lost ..( ok nevr mind i got! )

[@agirl2564]

mod

[@_Anna_Nass_]

Why did you write 2/8 for 2 divides 8 and 8/2 for 8 divides 2? That seems backwards to me. Is that how you do it for relations?

[@agirl2564]

it comes from the rule of symmetry.

[@KavyaKavya-xr2yb]

ill fail

[@elizebethfairway2130]

same :(