27-ClosuresEquivRels

# Each s i is called a block also called an equivalence

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Each S i is called a block (also called an equivalence class) Example: Suppose we form teams (e.g. for a doubles tennis tournament) from the set: {Abe, Kay, Jim, Nan, Pat, Zed} then teams could be: { {Abe, Nan}, {Kay, Jim}, {Pat, Zed} } Note: “on same team as” is reflexive, symmetric, transitive an equivalence relation. Equivalence relations and partitions are the same thing (two sides of the same coin).

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Discussion #27 Chapter 5, Sections 4.6-7 12/15 Partitions (continued…) Since individual elements can only appear in one block (S j S k = for j k), blocks can be represented by any element within the block. e.g. Nan’s Team John Stockon’s 1995 NBA finals team Formally, [x] = set of all elements related to x and y [x] iff xRy e.g. [Nan] represents {Abe, Nan}, Nan’s team [Abe] represents {Abe, Nan}, Abe’s team [John Stockton] represents the set of players who played in the playoffs for the Jazz in 1995
Discussion #27 Chapter 5, Sections 4.6-7 13/15 Partitions & Equivalence Relations The mod function partitions the natural numbers into equivalence classes. 0 mod 3 = 0 so 0 forms a class [0] 1 mod 3 = 1 so 1 forms new class [1] 2 mod 3 = 2 so 2 forms new class [2] 3 mod 3 = 0 so 3 belongs to [0] 4 mod 3 = 1 so 4 belongs to [1] 5 mod 3 = 2 so 5 belongs to [2] 6 mod 3 = 0 so 6 belongs to [0] Thus, the mod function partitions the natural numbers into equivalence classes. [0] = {0, 3, 6, …} [1] = {1, 4, 7,…} [2] = {2, 5, 8, …} Example:

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Discussion #27 Chapter 5, Sections 4.6-7 14/15 Partitions Equivalence Relations Theorem: If {S 1 , …, S n } is a partition of S, then R:S S is an equivalence relation, where R is “in same block as.” Note: to prove that R is an equivalence relation, we must prove that R is reflexive, symmetric, and transitive.
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• Winter '12
• MichaelGoodrich
• Equivalence relation, Binary relation, Transitive relation, Symmetric relation

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