If C 1,C 2 ∈ Pand C 1 6= C 2 then C … ⊆ is an order relation on sets. The relation R induces a partition on X. In the previous example, the suits are the equivalence classes. the class [x] is the inverse image of f(x).

Let X be a non-empty set. A partition of X is a collection of … Theorem 2. Equivalence relation and partitions An equivalence relation on a set Xis a relation which is reﬂexive, symmetric and transitive A partition of a set Xis a set Pof cells or blocks that are subsets of Xsuch that 1. Deﬁnition. Let R be an equivalence relation on a set A. If C∈ Pthen C6= ∅ 2. That is if x is a typical element of X, then “the equivalence class of x” is [x] = {t: tRx}. Problem 4.27: Let Q be the set of all rational numbers, and let R be the set of ordered pairs (x,y) in Q × Q such that when x and y are represented by fractions in lowest terms these fractions have the same denominator. Equivalence relation and partitions An equivalence relation on a set Xis a relation which is reﬂexive, symmetric and transitive A partition of a set Xis a set Pof cells or blocks that are subsets of Xsuch that 1. Let x,y,z ∈ Q. Let R be an equivalence relation on X . Proof. Also, when we specify just one set, such as \(a\sim b\) is a relation on set \(B\), that means the domain & codomain are both set \(B\). Theorem: An equivalence relation ∼ {\displaystyle \sim } on X {\displaystyle X} induces a unique partition of X {\displaystyle X}, and likewise, a partition induces a unique equivalence relation on X {\displaystyle X}, such that these are equivalent.

Theorem 1. It is to be noted that each [x] R is simply a subset of X.

An equivalence relation on X gives rise to a partition of X into equivalence classes. An order (or partial order) is a relation that is antisymmetric and transitive. Suppose R is an equivalence relation on a set A. A relation on a set \(A\) is an equivalence relation if it is reflexive, symmetric, and transitive. Corollary.

The partition it induces consists of one equivalence class, {Ann,Bob,Chip} (exhaustive and trivially disjoint). Theorem 7.1. Proof. Definition : Given an equivalence relation on a set, any element of lying in a given equivalence class is called a representative of that equivalence class. Now , so . An example of a partition of the integers under the relation of “having the same remainder when divided by 3.” Let [x] R be the equivalence class containing x∈X. Examples: ≤ is an order relation on numbers. Conversely, a partition of X gives rise to an equivalence relation on X whose equivalence classes are exactly the elements of the partition. Proof. Then there exist some integers m,n,p,q,j,k The set of all equivalence classes of elements of A is called A modulo R and is denoted A/R A/RxR x ?A 4 We are going to prove that any equivalence relation R on A induces a partition of A and any partition of A gives rise to an equivalence relation. (a) Prove that R is an equivalence relation on Q. Suppose is an equivalence relation on X. Then the equivalence classes of R form a partition of A. Then every partition of X induces an equivalence relation on X , and every equivalence relation induces a partition of X . Lemma Let A be a set and R an equivalence relation on A. Let X be a set. Clearly, . In general, if ∼ is an equivalence relation on a set X and x∈ X, the equivalence class of xconsists of all the elements of X which are equivalent to x. This equivalence relation is known as the kernel of f. More generally, a function may map equivalent arguments (under an equivalence relation ~ X on X) to equivalent values (under an equivalence relation ~ Y on Y). If C 1,C 2 ∈ Pand C 1 6= C 2 then C 1 ∩C 2 = ∅ 3. We often use the tilde notation \(a\sim b\) to denote a relation. This world's "likes" is also an equivalence relation. Order relations. If , let Thus, is the equivalence class of x. If a∈ Xthere exists C∈ Psuch that a∈ C 1. The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. Let X be a set. If C∈ Pthen C6= ∅ 2. Let P be the collection of distinct equivalence classes of X wrt R : P = f[x ] : x 2 X g: For any a A we define the equivalence class of a, written [a], by [a] = { x A : x R a}. 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.