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15a-Orders-and-Equivalences

15a-Orders-and-Equivalences - IOrders and Equivalences...

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Orders and Equivalences Discrete Mathematics
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Discrete Mathematics – Orders and Equivalences 15-2 Previous Lecture Properties of binary relations Equivalence relations reflexivity symmetricity transitivity anti-symmetricity
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Discrete Mathematics – Orders and Equivalences 15-3 Properties of binary relations Reflexivity A binary relation R A × A is said to be reflexive if (a,a) R for all a A. Symmetricity A binary relation R A × A is said to be symmetric if, for any a,b A, if (a,b) R then (b,a) R. Transitivity A binary relation R A × A is said to be transitive if, for any a,b,c A, if (a,b) R and (b,c) R then (a,c) R. Anti-symmetricity A binary relation R A × A is said to be anti-symmetric if, for any a,b A, if (a,b) R and (b,a) R then a =b.
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Discrete Mathematics – Orders and Equivalences 15-4 Equivalence relations A binary relation R on a set A is said to be an equivalence relations if it is reflexive, symmetric, and transitive. Let R People × People. Pair (a,b) R if and only if a and b are of the same age. Equivalence classes. Take a A. The set C(a) = { b | (a,b) R} is called the equivalence class of a. For example, C(my father) is the set of all 68 year old people. Let S Animals × Animals. Pair (a,b) S if and only if a and b belong to the same species.
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Discrete Mathematics – Orders and Equivalences 15-5 Equivalence Classes Lemma .
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