lecnotes3.3_1

# lecnotes3.3_1 - 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 } .

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