relations. binary relations a relation between elements of two sets is a subset of their cartesian...
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Relations
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Binary Relations
• a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs).
• Note the difference between a relation and a function: in a relation, each a ∈ A can map to multiple elements in B. Thus, relations are generalizations of functions.
• If an ordered pair (a, b) ∈ R then we say that a is related to b. We may also use the notation aRb.
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Relations
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Relations (Graph View)
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Relations (on a set)
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Reflexivity
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Symmetry I
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Symmetry II
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Symmetric Relations
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Transitivity
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Transitivity
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Combining Relations
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Composition and Powers
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Power Examples
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Equivalence Relations
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Equivalence Classes I
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Equivalence Classes II
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Partitions I
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Partitions II
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Matrix Interpretation
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Equivalence Relations (Example-I)
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Equivalence Relations (Example-II)
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Equivalence Relations (Example-III)
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Equivalence Relations (Example-IV)
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Partial Order I
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Partial Order II
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Equivalence Relations (Example-IV)
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Definition
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Comparability
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Total Orders
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Hasse Diagram
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Hasse Diagram Example