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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)}

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Equivalence Relations

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  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

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