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() The operation of subtraction A, BHA-B of matrices A, Bhaving the same size is defined by as follows: if A = |||| and B = ||by||,then A-B= ||aij – by|l- For instance, G :)-(: ) - (1–4 2-3) 3-2 4-1, Like other matrix operations, the operation of subtraction is very helpful when dealing with relations: it often enables one to get a clearer picture, to make surer conclusions, to avoid mistakes, etc. One of the possible applications is as follows. Let S,T be relations on a finite set A and let Ms and My be the matrices of S and T, respectively. Then the number of negative entries of the matrix M7 - Ms is equal to the cardinality of the set . (please enter an appropriate expression involving the sets S and T; use the keywords union, intersection, and setminus for the set-theoretic operations n, U, and \, respectively, if necessary) In particular, all entries of the matrix MT – Ms are nonnegative if and only if the relation S is . (enter a correct word) in the relation T. (i) Consider the relation R= {(1,1), (1,3), (1, 4), (1,5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3,1), (3, 3), (3, 4), (3,5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 5)} on the set A = {1,2, 3, 4, 5}. Verify whether or not the relation Ris transitive by realizing the following plan. (a) Find the matrix MRof the relation R (here, and below in (b,c), please enter the matrix row-by-row in the five input fields below; when entering each row, separate the entries by single spaces):

() The operation of subtraction A, BHA-B of matrices A, Bhaving the same size is defined by as follows: if A = |||| and B = ||by||,then A-B= ||aij – by|l- For instance, G :)-(: ) - (1–4 2-3) 3-2 4-1, Like other matrix operations, the operation of subtraction is very helpful when dealing with relations: it often enables one to get a clearer picture, to make surer conclusions, to avoid mistakes, etc. One of the possible applications is as follows. Let S,T be relations on a finite set A and let Ms and My be the matrices of S and T, respectively. Then the number of negative entries of the matrix M7 - Ms is equal to the cardinality of the set . (please enter an appropriate expression involving the sets S and T; use the keywords union, intersection, and setminus for the set-theoretic operations n, U, and \, respectively, if necessary) In particular, all entries of the matrix MT – Ms are nonnegative if and only if the relation S is . (enter a correct word) in the relation T. (i) Consider the relation R= {(1,1), (1,3), (1, 4), (1,5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3,1), (3, 3), (3, 4), (3,5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 5)} on the set A = {1,2, 3, 4, 5}. Verify whether or not the relation Ris transitive by realizing the following plan. (a) Find the matrix MRof the relation R (here, and below in (b,c), please enter the matrix row-by-row in the five input fields below; when entering each row, separate the entries by single spaces):

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discrete Mathematics

) The operation of subtraction А, ВНА-В of matrices A, Bhaving the same size is defined by as follows: if A = ||aij|| and B = ||b|l, then A - B= ||aj – bij|l- For instance, (: )-(: )-()-(G :) (1-4 2-3 3 –2 4-1 3 Like other matrix operations, the operation of subtraction is very helpful when dealing with relations: it often enables one to get a clearer picture, to make surer conclusions, to avoid mistakes, etc. One of the possible applications is as follows. Let S, T be relations on a finite set A and let Ms and MT be the matrices of S and T, respectively. Then the number of negative entries of the matrix Mr - Ms is equal to the cardinality of the set. (please enter an appropriate expression involving the sets S and T; use the keywords union, intersection, and setminus for the set-theoretic operations n, U, and \, respectively, if necessary) In particular, all entries of the matrix MT – Ms are nonnegative if and only if the relation S is . (enter a correct word) in the relation T. (ii) Consider the relation R= {(1,1), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3,1), (3, 3), (3, 4), (3, 5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 5)} on the set A = {1,2, 3, 4, 5}. Verify whether or not the relation Ris transitive by realizing the following plan. (a) Find the matrix MR of the relation R (here, and below in (b,c), please enter the matrix row-by-row in the five input fields below; when entering each row, separate the entries by single spaces):
Transcribed Image Text:) The operation of subtraction А, ВНА-В of matrices A, Bhaving the same size is defined by as follows: if A = ||aij|| and B = ||b|l, then A - B= ||aj – bij|l- For instance, (: )-(: )-()-(G :) (1-4 2-3 3 –2 4-1 3 Like other matrix operations, the operation of subtraction is very helpful when dealing with relations: it often enables one to get a clearer picture, to make surer conclusions, to avoid mistakes, etc. One of the possible applications is as follows. Let S, T be relations on a finite set A and let Ms and MT be the matrices of S and T, respectively. Then the number of negative entries of the matrix Mr - Ms is equal to the cardinality of the set. (please enter an appropriate expression involving the sets S and T; use the keywords union, intersection, and setminus for the set-theoretic operations n, U, and \, respectively, if necessary) In particular, all entries of the matrix MT – Ms are nonnegative if and only if the relation S is . (enter a correct word) in the relation T. (ii) Consider the relation R= {(1,1), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3,1), (3, 3), (3, 4), (3, 5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 5)} on the set A = {1,2, 3, 4, 5}. Verify whether or not the relation Ris transitive by realizing the following plan. (a) Find the matrix MR of the relation R (here, and below in (b,c), please enter the matrix row-by-row in the five input fields below; when entering each row, separate the entries by single spaces):
(b) Find the matrix M2 of the relation R2: (c) Now find-for your own use-the matrix MR- Me. This matrix has . (please enter a suitable number) negative entries, which means that, |R \R = %3D (d) Accordingly, by (1) and by (c), the relation R . (enter two correct words, or three correct words) in the relation R, and hence the relation R is whie one i choose O is not transitive.
Transcribed Image Text:(b) Find the matrix M2 of the relation R2: (c) Now find-for your own use-the matrix MR- Me. This matrix has . (please enter a suitable number) negative entries, which means that, |R \R = %3D (d) Accordingly, by (1) and by (c), the relation R . (enter two correct words, or three correct words) in the relation R, and hence the relation R is whie one i choose O is not transitive.
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