Relations
This text delves into the concept of binary relations, defining them as subsets of the Cartesian product of two sets. Unlike functions, relations allow for multiple mappings from one set to another. Key properties of relations such as reflexivity, symmetry, and transitivity are explored, alongside their implications in equivalence relations and partial orders. Additionally, examples illustrate equivalence classes and total orders, with a focus on Hasse diagrams. This comprehensive overview aids in grasping the foundational aspects of relations in mathematics.
Relations
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Presentation Transcript
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.
