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Lec08_Relations

# Lec08_Relations - TDS1191 Discrete Structures Lecture 08...

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Discrete Structures 1 TDS1191 Discrete Structures Lecture 08 Relations Multimedia University Trimester 2 Session 2011/2012 [Lecture08][Relations]

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Discrete Structures 2 [Introduction] Binary Relations Composition of Relations Equivalence Relation Partially Ordered Sets Hasse Diagrams [Lecture08][Relations]
Discrete Structures 3 From sets to ordered pairs Lets define: (a, b) = df {{a}, {a,b}} This gives, {5, 8} = {8, 5} (5, 8) (8, 5) [Lecture08][Relations] ALERT!!!! {1,2} ≠ (1,2)

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Discrete Structures 4 We can explicitly name empty sequences, singlets, pairs, triples, quadruples, quintuples, …, n-tuples. Are these n-tuple (1, 2) , (2, 1) , (2, 1, 1) equivalent? Ordered n-tuples Generalizing the definition of ordered pairs we define (a 1 , a 2 , …, a n ) to be an ordered n-tuple (or a sequence of length n). [Lecture08][Relations]
Discrete Structures 5 A × B = df {(a, b) | a A b B } is called the Cartesian product of A and B. Cartesian Product {a, b} × {1, 2} = {(a, 1), (a, 2), (b, 1), (b, 2)} Take note, For finite sets A, B: |A × B| = |A| |B|. The Cartesian product is not commutative: ¬ ( 2200 A 2200 B: A × B = B × A). The Cartesian product can be easily extended for n sets as A 1 × A 2 × × A n For any set A, A × = = × A Give sets A and B where A × B B × A. Give {1,2} × {1,2,3} × {c,d}. [Lecture08][Relations] Try to draw a diagram to see how Cartesian product works

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Discrete Structures 6 R A 1 × … × A n is called n-ary relation R on the sets A 1 ,…,A n. The sets A i are called the domains of R. The degree or arity of R is n. For n = 2, we say it is a binary relation. aRb, (a, b) R are writings for stating that a relates to b under R. R −1 = {(b, a) | (a, b) R} is called the inverse relation of R A binary relation R from a set A into a set B is a subset of A ×B A binary relation R from a set A into itself is also called a relation on A.
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