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lecnotes3.3.2

# lecnotes3.3.2 - Lecture Notes Concepts of...

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Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation A–D, Spring 2008 3 Relations 3.3 Relations, Equivalence Relations, and Partitions Recall the definition of the Cartesian product X × Y of two sets X and Y . From basic set theory, we have: Theorem 3.1 If A , B , C , and D are sets, then (a) A × ( B C ) = ( A × B ) ( A × C ) (b) A × ( B C ) = ( A × B ) ( A × C ) (c) A × ∅ = (d) ( A × B ) ( C × D ) = ( A C ) × ( B D ) (e) ( A × B ) ( C × D ) ( A C ) × ( B D ) Proof: (Part (a)) The ordered pair ( x, y ) A × ( B C ) iff x A and y B C iff x A and ( y B or y C ) iff ( x A and x B ) or ( x A and y C ) iff ( x, y ) A × B or ( x, y ) A × C iff ( x, y ) ( A × B ) ( A × C ). Therefore A × ( B C ) = ( A × B ) ( A × C ). The other parts of the proof are left as an exercise. Definition 3.1 A relation R from a set X to a set Y is a subset of X × Y : R = { ( x, y ) X × Y | xRy } . Subsets of X × X are called relations on X . Definition 3.2 The domain of the relation R from X to Y is the set Dom ( R ) = { x X | there exists y Y such that xRy } . The range of the relation R from X to Y is the set Rng ( R ) = { y Y | there exists x X such that xRy } . 1

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A relation R can be defined by listing the elements of R or by describing the property that relates the two elements. Example 3.1 Let A = { 1 , 2 , 3 } and B = { a, 5 , { b } , c } and R be given by R = { (1 , a ) , (1 , { b } ) , (2 , a ) , (2 , c ) } . Then we have 1 Ra , 1 R { b } , 2 Ra , 2 Rc , 3 R 5 Dom ( R ) = { 1 , 2 } ⊆ A Rng ( R ) = { a, { b } , c } ⊆ B Example 3.2 Let P be the set of all people. Let L = { ( a, b ) P × P | a and b have the same last name } . Then L is a relation on P . Example 3.3 Let S be a relation on the set N × N defined by ( m, n ) S ( k, j ) iff m + n = k + j . Then (3 , 17) S (12 , 8) but (5 , 4) S (6 , 15) . Note that we can write S as S = { (( m, n ) , ( k, j )) N 2 × N 2 | m + n = k + j } . What are the domain and range of S ? Relations that are defined on familiar sets can be expressed visually by using a graph . For example, let S = { ( x, y ) R × R | x 2 + y 2 1 } and consider its graph (draw graph here). Now consider the set A = { 3 , 6 , 9 , 12 } and let R on A be defined by, for x, y A , xRy iff x divides y . Not only can this be represented with a graph, it can also be represented with a directed graph or digraph . We think of objects in A as vertices and think of R as telling us which vertices are connected by edges. The edges are directed from one vertex to another using arrows. There is an edge from x to
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lecnotes3.3.2 - Lecture Notes Concepts of...

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