1 / 13

Equivalence Relations

Equivalence Relations. A relation on set A is called an equivalence relation if it is: reflexive, symmetric, and Transitive We use the notation a  b. Example. Let R be a relation on set A , where A = {1, 2, 3, 4, 5} and R = {(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1)}

azizi
Télécharger la présentation

Equivalence Relations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Equivalence Relations • A relation on set A is called an equivalence relation if it is: • reflexive, • symmetric, and • Transitive • We use the notation a b

  2. Example • Let R be a relation on set A, where A = {1, 2, 3, 4, 5} and R = {(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1)} • Is R an equivalence relation?

  3. Another example: Let A={set of lines } and R is a relation R={(L1,L2):L1 is parallel to L2}

  4. Representing Relations Using Matrices • Let R be a relation from A to B A = {a1, a2, … } B = {b1, b2, … } • The zero-one matrix representing the relation R has a 1 as its entry when ai is related to bj and a 0 in this position if ai is not related to bj.

  5. Example • Let R be a relation from A to B A={a, b, c} B={d, e} R={(a, d), (b, e), (c, d)} • Find the relation matrix for R

  6. Relation Matrix • Let R be a relation from A to B • A={a, b, c} • B={d, e} • R={(a, d), (b, e), (c, d)} Note that A is represented by the rows and B by the columns in the matrix. Cellij in the matrix contains a 1 iff ai is related to bj.

  7. Relation Matrices and Properties • Let R be a binary relation on a set A and let M be the zero-one matrix for R. • R is reflexive iff Mii = 1 for all i • R is symmetric iff M is a symmetric matrix

  8. Relation Matrices and Properties

  9. 1 1 0 é ù ê ú = M 1 1 1 ê ú R ê ú 0 1 1 ë û Example • Suppose that the relation R on a set is represented by the matrix MR. • Is R reflexive and symmetric?

  10. 1 1 0 é ù ê ú = M 1 1 1 ê ú R ê ú 0 1 1 ë û Example Is R reflexive? Is R symmetric? • All the diagonal elements = 1, so R is reflexive. • The lower left triangle of the matrix = the upper right triangle, so R is symmetric.

  11. Let A={a1,a2,a3} and B={b1,b2,b3,b4,b5} which ordered pairs are in the relation R represented by the matrix ?? 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1

  12. intersection union

  13. Find the union and intersection of zero_one matrices The union The intersection

More Related