lecnotes3.3_1 - Lecture Notes, Concepts of Mathematics...

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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation AD, 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 ....
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lecnotes3.3_1 - Lecture Notes, Concepts of Mathematics...

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