lect_35 - Equivalence Relations and Partitions Margaret M...

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Unformatted text preview: Equivalence Relations and Partitions Margaret M. Fleck 23 April 2010 This lecture continues the topic of equivalence relations (section 8.5 of Rosen), including how to prove that something is an equivalence relation. 1 Announcements Reminder: quiz next Wednesday. 2 Partitions Equivalence relations are used to divide up a set A into equivalence classes, each of which can then be treated as a single object. E.g. all the equivalence fractions are treated as a single rational number. This only works properly if the equivalence classes divide up the set neatly: they cover the whole set A , they don’t overlap, and every equivalence class contains at least one representative element. Cutting up A neatly is called being a partition of A . Knowing that we have a partition means that we don’t have to worry about how to handle annoying situations like partial overlap between two equivalence classes. This is particularly important when defining operations such as addition (e.g. of rationals). How would we define addition if one of the input equivalence classes had nothing in it? Formally, a partition of a set A is a collection of non-empty subsets of 1 A which cover all of A and don’t overlap. Specifically, if the subsets are A 1 , A 2 , . . .A n , then they must satisfy three conditions: a) A 1 ∪ A 2 ∪ . . . ∪ A n = A b) A i negationslash = ∅ for all i c) A i ∩ A j = ∅ for all i negationslash = j . A partition can contain an infinite set of subsets. To cover this possibility, we need to use a more general notation. Let P be our partition. then the three conditions are: a) uniontext X ∈ P X = A b) X negationslash = ∅ for all X ∈ P c) X ∩ Y = ∅ for all X, Y ∈ P , X negationslash = Y 3 Need for the RST properties Any partition P has a corresponding equivalence relation. Specifically, we define x ∼ y...
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lect_35 - Equivalence Relations and Partitions Margaret M...

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