Equivalence relations - Columbia University
equivalence relation, provided we restrict to a set of sets (we cannot just de ne this as an equivalence relation on the \set" of all sets, since this is too big to be a set) For example, we could de ne this relation on a set such as P(R), the set of all subsets of the real numbers The
EquivalenceRelations
These three properties are captured in the axioms for an equivalence relation Definition An equivalence relation on a set X is a relation ∼ on X such that: 1 x∼ xfor all x∈ X (The relation is reflexive ) 2 If x∼ y, then y∼ x (The relation is symmetric ) 3 If x∼ yand y∼ z, then x∼ z (The relation is transitive ) Example
95 Equivalence Relations
Corollary If A is a set, R is an equivalence relation on A, and a and b are elements of A, then either [a] \[b] = ;or [a] = [b]: That is, any two equivalence classes of an equivalence relation are either mutually disjoint or identical Theorem 2 Let R be an equivalence relation on a set A Then the equivalence classes of R form a partition of A
Equivalence Relations - Mathematical and Statistical Sciences
An Important Equivalence Relation Let S be the set of fractions: S ={p q: p,q∈ℤ,q≠0} Define a relation R on S by: a b R c d iff ad=bc This relation is an equivalence relation 1) For any fraction a/b, a/b R a/b since ab = ba (Reflexitivity) 2) If a/b R c/d, then ad = bc, so cb = da and c/d R a/b (Symmetry)
Equivalence Relations and Functions
Equivalence Relations and Functions October 15, 2013 Week 13-14 1 Equivalence Relation A relation on a set X is a subset of the Cartesian product X£X Whenever (x;y) 2 R we write xRy, and say that x is related to y by R
Math 127: Equivalence Relations
De nition 4 Let ˘be an equivalence relation on X The set [x] ˘as de ned in the proof of Theorem 1 is called the equivalence class, or simply class of x under ˘ We write X= ˘= f[x] ˘jx 2Xg Example 6 If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence classes: odds and evens
Section 42: Equivalence Relations
Then ~ is an equivalence relation because it is the kernel relation of function f:S N defined by f(x) = x mod n Example: Let x~y iff x+y is even over Z Note that x+y is even iff x and y are both even or both odd iff x mod 2 = y mod 2 Therefore ~ is an equivalence relation because ~ is the kernel relation of
QUIVALENCE ELATIONS
equivalence relations, partitions, and quotient sets are important Example 3 1 Define the following equivalence relation on Z (Znf0g): (a;b)˘(c;d),ad =bc: Then (Z Z)=˘“is” the rational numbers More about this later Example 3 2 Let ˘be the equivalence relation on R defined by x ˘y if and only if x y is an integer multiple of 2p
Daniel ALIBERT Ensembles, applications Relations d
Une relation réflexive, symétrique et transitive est appelée une relation d'équivalence Définition Soit E un ensemble, muni d'une relation d'équivalence R Pour tout élément x de E, on appelle classe d'équivalence de x et l'on note C(x) le sous-ensemble de E formé des éléments y tels que x R y soit vrai
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