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13h - 13-2 Discrete Mathematics Relations Previous Lecture...

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Introduction Relations Discrete Mathematics Andrei Bulatov Discrete Mathematics - Relations 13-2 Previous Lecture Venn diagrams Operations of Connection to logic Laws of set theory s Intersection s Union s Symmetric difference s Complement s Difference Discrete Mathematics - Relations 13-3 Relations ` Relation ’, the connection between things or people Between people, family relations `to be brothers’ x is a brother of y `to be older’ x is older than y `to be parents’ x and y are parents of z Between things, numerical relations `to be greater than’ x < y on the set of real numbers `to be divisible by’ x is divisible by y on the set of integers Between things and people, legal relations `to be an owner’ x is an owner of y Discrete Mathematics - Relations 13-4 Cartesian Product The Cartesian product of sets A and B, denoted by A × B, is the set of all ordered pairs of elements from A and B. A × B = { (a,b) | a A, b B } The elements of the Cartesian product are ordered pairs. In particular, (a,b) = (c,d) if and only if a = c and b = d. If sets are thought of as `1-dimensional’ objects, Cartesian products are 2-dimensional 1 2 3 4 5 1 2 3 {1,2,3,4,5} × {1,2,3} 3 1 5 2 (2,5) × (1,3) 1 2 3 4 5 6 7 8 a b c d e f g h Discrete Mathematics - Relations 13-5 Cartesian Product of More Than Two Sets Instead of ordered pairs we may consider ordered triples , or, more general, k-tuples . (a,b,c),
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13h - 13-2 Discrete Mathematics Relations Previous Lecture...

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