Equivalence Relations

We look at the notion of an equivalence relation on a set, define an equivalence class, and consider several examples.

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[goodplacetostop]

13:41 Pay great attention, Michael’s students 😛
19:41 Good Place To Stop

Homework :
- Give an equivalence relation • on R such that the equivalence class of any element is size 4.
- Give an equivalence relation • on R such that there are infinitely many equivalence classes of R under •, and each equivalence class has the same cardinality as R.

[tomkerruish]

I like replacing symmetry and transitivity with euclideanity (euclideanness?), viz., xRy and xRz together imply yRz. This, along with reflexivity, imply symmetry and thence transitivity.

[kevinmartin]

It should be noted that all the equivalence classes are disjoint (the transitivity prevents them from sharing elements of A) and, taken together, span A×A (because the reflexiveness applies to *all* elements of A).

This further means that the sum of the sizes of all the equivalence classes equals |A|. The total number of distinct equivalence relations on A ends up being the number of partitions of |A|.

[kevinmartin]

The examples would have been more interesting if A had included -3 and 3 and "same sign" were "same absolute value" instead. "Same parity" produces two equivalence classes each containing half the elements of A, whereas "same absolute value" would produce |A|/2 equivalence classes each containing two elements.

[tomatrix]

Another Video:) How great. Good work Michael

[MrYesman]

Is there a playlist with the videos from this course?

[nontth]

Is this some kind of set theory?? We don't learn this kind of thing in highschool are we? But this seems interesting. Where can I find more info on this topic?

[debjitmullick]

Sir, how can we join the 'proof writing' skill program ...??? Can we ...??

[rektator]

Every equivalence relation on a set is induced via a function in the following sense: Let f:X->Y be a function between sets X and Y. We define an equivalence relation R on X by setting that xRy iff f(x) = f(y) for elements x and y in X. This is clearly an equivalence relation and it is called the set theoretic kernel of function f and often denoted ker f (In algebraic models, these kernels become very important)

Suppose that there exists an equivalence relation R on X. Consider the function p:X->X/R, p(x) = [x] for elements x in X. Now R = ker p. Hence every equivalence relation is obtained from a function using equality.

[pedroalonso]

I enjoyed this video a lot but I missed a mention to partitions. I guess they will be explained in a next video.

[flux]

Interesting integral question, Integrate arctan(x)arcos(x) from 0 to 1, like so Michael sees this

[MathCuriousity]

Hey good kind being. 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!!

[angel-ig]

11:56 "So the equivalence class of the empty set is just the empty set": wrong. The correct thing is "So the equivalence class of the empty set is the set containing just the empty set"

[shanmukeshr]

I hate this topic sometimes it makes no sense