1 / 21

Section 7.3

Section 7.3. Representing relations (part 1: matrices). Representing relations. We have already seen that relations between finite sets can be represented using lists of ordered pairs To represent relations in computer programs, zero-one matrices are usually employed

ezra
Download Presentation

Section 7.3

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. Section 7.3 Representing relations (part 1: matrices)

  2. Representing relations • We have already seen that relations between finite sets can be represented using lists of ordered pairs • To represent relations in computer programs, zero-one matrices are usually employed • Directed graphs provide pictorial representations of relations, and are useful in determining some of their properties

  3. Matrices • A matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns • Each entry in a matrix can be designated using subscripts representing the entry’s row and column: the element at row i, column j of matrix A would be designated aij • The entire matrix A is often represented with the abbreviation [aij]

  4. Square matrices • The size of a matrix is usually represented as m x n, with m representing the number of rows and n the number of columns • If m=n, then matrix A is a square matrix

  5. Examples The matrix at the right is a 4x3 matrix; the values in row 1 are 1 (at column 1), through 4 (at column 4). The value of a23 is 7. In the square matrix at the right, the elements a11, a22 and a33 form the main diagonal of the matrix.

  6. Transpose of a matrix • If A = [aij] is an m x n matrix, then the transpose of A is the n x m matrix AT = [aTij] • We can obtain AT by interchanging the rows and columns of A • The transpose of a transpose is the original matix: (AT)T = A

  7. Boolean matrix • A zero-one, or Boolean matrix, is a matrix consisting of binary digits • Such a matrix can be used to represent a relation between finite sets • A relation R from set A to set B can be represented by MR = [mij] where • mij = 1 if (ai, bj)  R • mij = 0 if (ai, bj)  R

  8. Example Suppose set A = {a, b, c, d} and set B = {a, c, d} and relations from A to B R1 = {(a,a), (a,c), (b,d), (d,d)} and R2 = {(a,c), (b,d), (c,a), (c,d), (d,a), (d,c)}

  9. Operations on Boolean matrices • Suppose A=[aij] and B=[bij] are Boolean matrices and are the same size • The join of A and B is matrix C=[cij], represented by A V B: • cij = 1 if aij = 1 or bij =1 • cij = 0 if both aij and bij are 0 • The meet of A and B is matrix D=[dij], represented by A ^ B: • dij = 1 if aij = bij = 1 • dij = 0 if either aij or bij = 0

  10. Join and meet: example

  11. Boolean product of matrices • The Boolean product of m x p matrix A and p x n matrix B, denoted A B, is the m x n matrix C = [cij] defined by: • cij = 1 if aik = 1 and bkj =1 for some k between 1 and p • cij = 0 otherwise

  12. Finding the Boolean product • For any i and j, cij C = A B can be computed as follows: • Select row i of A and column j of B and arrange them side by side • Compare corresponding entries; if even a single pair of corresponding entries consists of 2 1’s, then cij = 1 – if not, then cij = 0

  13. Finding Boolean product

  14. Example

  15. Matrix of a relation on a set • … is a square matrix • … can be used to determine certain properties of the relation

  16. Reflexivity • A relation R on set A is reflexive if (a,a)R whenever a  A • So R is reflexive if m11=1, m22=1 … mnn=1 • In other words, R is reflexive if all elements on the main diagonal of MR = 1

  17. Symmetry • Any relation R is symmetric if (a,b)R implies that (b,a)R • A relation R on set A = [aij] is symmetric if and only if (aj,ai)R whenever (ai,aj)R • In matrix MR, we can determine R is symmetric if and only if mij = mji for all pairs i, j, with i = 1 … n and j = 1 … n • In other words, R is symmetric if and only if MR = (MR)T

  18. Antisymmetry • A relation R is antisymmetric if and only if (a,b)R and (b,a)R imply that a=b • A matrix of an antisymmetric relation has the property that: • If mij = 1, then mji = 0 if ij • Or, if ij, then either mij or mji = 0 (the other could be 0 or 1)

  19. Example …see the blackboard; see also exercise 7, p 495

  20. Representing unions and intersections • The Boolean matrix operations join and meet can be used to find matrices representing the union and intersection of two relations • Suppose R1 and R2 are relations on set A; then they are represented by matrices MR1 and MR2 • The matrix representing the intersection of R1 and R2 has a 1 in the positions where both MR1 and MR2 have a 1; the matrix representing their union would have a 1 where either has a 1: MR1 R2 = MR1  MR2 MR1 R2 = MR1  MR2

  21. Example See, again, the blackboard …

More Related