Equivalence Relations - Reflexive, Symmetric, and Transitive 

bruh its a illusion board its called light board

I just noticed.

Shouldn't it be: if y≥ x + 1 and x + 1 ≥ z, then y ≥ z ?

This is a stumbling block for me as well. Seems arbitrary and esoteric

Thanks!

Can I ask you some questions?? Plz ☺️

Love. I may love my wife, but that doesn't mean she loves me. Not symmetric.

relations are defined on cartesian products like Z^2 or R^2, not on rational numbers like 6/2. What do you mean by 'the equality relation of the object 3 over itself but of 3 on 6/2'? I think you may need to study the definition of a relation more

@@trigfunction ok here is my question what is the structure being preserved with homomorphisms? Can you give me an example or better two? I also don’t understand why a function that takes affine space to affine space is not a homomorphism.

It seems like you are trying to represent an equivalence relation within set theory. The idea of using a subset of the Cartesian product of elements a and b, mapped to a set containing true or false, is on the right track, but there are some details that need clarification. An equivalence relation � R on a set � A typically satisfies three properties for all elements � , � , � a,b,c in � A: Reflexivity: � � � aRa for all � a in � A. Symmetry: If � � � aRb, then � � � bRa. Transitivity: If � � � aRb and � � � bRc, then � � � aRc. Now, let's try to represent this using set theory. The idea is to represent the relation � R as a set of ordered pairs satisfying these properties. A common way to represent an equivalence relation is using equivalence classes. An equivalence class [ � ] [a] of an element � a is the set of all elements related to � a under � R. In set-builder notation, it can be written as [ � ] = { � ∈ � ∣ � � � } [a]={b∈A∣aRb}. To represent the equivalence relation � R as a set, you can use the following approach: Define the set: Let � A be the set on which the equivalence relation is defined. Define the relation set � T: � = { ( � , � ) ∈ � × � ∣ � � � } T={(a,b)∈A×A∣aRb} Define the equivalence class set � U: � = { [ � ] ∣ � ∈ � } U={[a]∣a∈A} where [ � ] = { � ∈ � ∣ � � � } [a]={b∈A∣aRb}. So, � T is the set of ordered pairs related under � R, and � U is the set of equivalence classes. Remember that in set theory, you're dealing with sets and their elements, and the ordered pairs in � T represent elements related under � R. The elements of � U are sets of elements related to each other, forming equivalence classes. The mapping to true or false might not be necessary in this context, as the focus is on capturing the relationships between elements in the equivalence classes.

It seems like you are trying to represent an equivalence relation within set theory. The idea of using a subset of the Cartesian product of elements a and b, mapped to a set containing true or false, is on the right track, but there are some details that need clarification. An equivalence relation � R on a set � A typically satisfies three properties for all elements � , � , � a,b,c in � A: Reflexivity: � � � aRa for all � a in � A. Symmetry: If � � � aRb, then � � � bRa. Transitivity: If � � � aRb and � � � bRc, then � � � aRc. Now, let's try to represent this using set theory. The idea is to represent the relation � R as a set of ordered pairs satisfying these properties. A common way to represent an equivalence relation is using equivalence classes. An equivalence class [ � ] [a] of an element � a is the set of all elements related to � a under � R. In set-builder notation, it can be written as [ � ] = { � ∈ � ∣ � � � } [a]={b∈A∣aRb}. To represent the equivalence relation � R as a set, you can use the following approach: Define the set: Let � A be the set on which the equivalence relation is defined. Define the relation set � T: � = { ( � , � ) ∈ � × � ∣ � � � } T={(a,b)∈A×A∣aRb} Define the equivalence class set � U: � = { [ � ] ∣ � ∈ � } U={[a]∣a∈A} where [ � ] = { � ∈ � ∣ � � � } [a]={b∈A∣aRb}. So, � T is the set of ordered pairs related under � R, and � U is the set of equivalence classes. Remember that in set theory, you're dealing with sets and their elements, and the ordered pairs in � T represent elements related under � R. The elements of � U are sets of elements related to each other, forming equivalence classes. The mapping to true or false might not be necessary in this context, as the focus is on capturing the relationships between elements in the equivalence classes.

@@suyash8264 Hey that was AMAZING! My follow up question is more--deep and fundamental! Hopefully you can grasp my question: inspired my whole wish to figure out if set theory can within itself have relations which map certain a set of propositions (for instance the proposition “set A is a subset of Set B) to to a set of truth values (true or false). So in your opinion, what’s the big problem with what I want to do? Basically use relations in set theory to state a proposition is true or false? Or perhaps I’m asking too much of this mapping? Meaning I’m assuming the mapping means “this proposition is true” which is on a meta level actually and Not what the mapping of some proposition to “true” is actually saying?!!!

@@suyash8264 chatgpt much?

@@MathCuriousity In classical logic and set theory, propositions can be encoded as sets, and truth values true or false can be represented by the sets {1} and {}, respectively. For example, you might represent the proposition "Set A is a subset of Set B" as a set, and if this set is non-empty, it is considered true; otherwise, it is false. One potential challenge you might encounter is the issue of self-reference and paradoxes. Set theory, particularly when dealing with propositions about sets (like Russell's paradox), can lead to logical contradictions. For instance, the set of all sets that do not contain themselves leads to a paradox. In formalizing propositions within set theory, Godel's incompleteness theorems also come into play. These theorems suggest that there will always be true mathematical statements that cannot be proven within a given formal system, introducing limitations on the completeness and consistency of such systems. Additionally, the level of abstraction involved in dealing with propositions about sets might make it challenging to directly map all propositions to truth values within the set theory framework. Some propositions may refer to concepts outside the formal system, leading to potential limitations. while using set theory to encode propositions and truth values is a well established and valuable approach in mathematical logic, it's essential to be aware of potential pitfalls related to self reference, paradoxes, and the limits imposed by Godel's incompleteness theorems.

Well sort of. The point is that the symbol = does mean something different in modular arithemetic than it does with normal integers. But nevertheless, those differences still make it an equivalence relation which is why we feel embolded to just call it "equality" again. Otherwise, we might want to use a different symbol and mean something different by it.

gg no re

@@DeezNardsX I somehow got a b in this class. Couldn’t tell you how

@@lukemiller867 pray for me I have my final in 3 weeks

@@DeezNardsX good luck bro that final was so hard

He just flipped the video

Flipped the video while editing

okay my bad, did some research and %3 is a relation itself, thus i correct myself, a%3 = a%3, reflexivity a%3 = b%3 => b%3 = a%3, symmetry if a%3 = b%3 and b%3 = c%3 then c%3 = a%3