Equivalence classes are either equal or disjoint
From Maths
Stub grade: B
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
I'm sure I've already done this SOMEWHERE - find it!
Contents
[hide]Statement
Let X be a set, let ∼⊆X×X be an equivalence relation on X, let X∼ denote the quotient of X by ∼[Note 1] and lastly let π:X→X∼ given by π:x↦[x] be the canonical projection of the equivalence relation. Then:
- We claim that X∼ is a partition of X. That is:
- ∀x∈X∃y∈X∼[x∈y] - all elements of x belong to an element of the partition
- ∀u,v∈X∼[u∩v≠∅⟹u=v] - if u and v are not disjoint, they are equal
- Equivalently (by contrapositive[Note 2]): ∀u,v∈X∼[u≠v⟹u∩v=∅]
Proof
Grade: B
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
This proof has been marked as an page requiring an easy proof
The message provided is:
This is an easy and routine proof. First year friendly and all. However:
- Caution:I must make sure to prove the requirements e.g. that [x], the equivalence class containing x makes sense. As if I use a property like:
- y∈[x]⟺y∼x
- I'm sort of indirectly using part 2 of this theorem if I ever use the transitive property of equivalence relations involving x and y.
This proof has been marked as an page requiring an easy proof
Notes
- Jump up ↑ In other words: X∼ is the set of equivalence classes of ∼
- Jump up ↑ The contrapositive of A⟹B is (¬B)⟹(¬A). That is to say:
- (A⟹B)⟺((¬B)⟹(¬A))
References
Grade: D
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
Do not worry I know this content really well, this result is true. I promise
- I need to find a book that deems this theorem worthy of making explicit!
Categories:
- Stub pages
- Pages requiring proofs: Easy proofs
- Pages requiring proofs
- Pages requiring references
- Theorems
- Theorems, lemmas and corollaries
- Abstract Algebra Theorems
- Abstract Algebra Theorems, lemmas and corollaries
- Abstract Algebra
- Elementary Set Theory Theorems
- Elementary Set Theory Theorems, lemmas and corollaries
- Elementary Set Theory
- Set Theory Theorems
- Set Theory Theorems, lemmas and corollaries
- Set Theory