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
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
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
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
equivalence relation notes - People
Before we show that this gives an equivalence relation, it is a good idea to express the notion of being ”evenly divisible” in a way that is more accessible to us: we say that an integer x is evenly divisible by a non-zero integer d if
22 Equivalence Relations - NIU
The given relation is ∼f, which we know to be an equivalence relation 20 Let ube a fixed vector in R3, and assume that uhas length 1 For vectors vand w, define v∼ wif v·u= w·u, where · denotes the standard dot product Show that ∼ is an equivalence relation, and give a geometric description of the equivalence classes of ∼
Equivalence Relations - facksuedusa
DEFINITION 3 Let R be an equivalence relation on a set A The set of all elements that are related to an element a of A is called the equivalence class of a The equivalence class of a with respect to R is denoted by [a] [for this equivalence class ]= { ∈????∶ } Equivalence Classes and Partitions THEOREM 1 Let R be an equivalence relation
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