lect11 - Equivalence relations and partitions. Consider the...

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1 Equivalence relations and partitions . Definition. A relation R A × A is called an equivalence relation on A if it is symmetric, reflexive and transitive. Consider the following relation on a set of all people: B = {( x , y )| x has the same birthday as y } B is reflexive, symmetric and transitive. We can think about this relation as splitting all people into 366 categories, one for each possible day. An equivalence relation on a set A represents some partition of this set.
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2 Definition. For any set A subsets A i A partition set A if A = A 1 A 2 A n A i A j = , for any i j . A i for any i Example. A ={1, 2, 3, 4}, Π ={{2}, {1, 3}, {4}} is a partition of A . A 1 A 2 A 3 A 4 A
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3 Definition. Suppose R is equivalence relation on a set A, and x A . Then the equivalence class of x with respect to R is the set [ x ] R ={ y A | yRx } In the case of the same birthday relation B , if p is any person, then the equivalence class of p [ p ] B ={ q P | pBq } ={ q P | q has the same birthday as p } For example, if John was born on Aug. 10, [John] B = { q P | q was born on Aug.10} The set of all equivalence classes of elements of A is called A modulo R and is denoted A / R: A / R ={[ x ] R | x A }
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4 Theorem 1. Suppose R is an equivalence relation on a set A. Then A / R ={[ x ] R | x A } is a partition of A. 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. Proof. To prove that A / R defines a partition, we must prove three properties of a partition. 1) The union of all equivalence classes [ x ] R equals A , i. e. Since any equivalence class is a subset of A , their union is also a subset of A. So, all we need to show is To prove this, suppose x A. Then x [ x ] R because [ x ] R ={ y A | yRx }and xRx due to reflexive property of R. A x R A x = ] [ 1 R A x x A ] [
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5 2) To prove that A / R is pairwise disjoint we need to show, that for any x , y A if [ x ] [ y ] then [ x ] [ y ]= .
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lect11 - Equivalence relations and partitions. Consider the...

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