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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 } .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern