For each of the following sets B and binary relations ~, decide whether ~defines an equivalence relation on B. Justify your answer in each case.В — R(а)Relation: x ~ y if |x – y| < 1Set:B = Mn(R), the set of n xn matrices with realentries, where n is some fixed positive integerX ~Y if Tr(X – Y) e Z(ъ)Set:Relation:(Here, Tr(X) denotes the trace of a matrix X E Mn(R).)

Here we are given two st and we have to determine whether they define an equivalence relation…

For each of the following sets B and binary relations ~, decide whether defines an equivalence relation on B. Justify your answer in each case. (a) Relation: x y if |x – y| < 1 Set: B = R B = Mn(R), the set of n xn matrices with real entries, where n is some fixed positive integer X ~ Y if Tr(X – Y) e Z (b) Set: Relation: (Here, Tr(X) denotes the trace of a matrix X € Mn (R).)

For each of the following sets B and binary relations ~, decide whether defines an equivalence relation on B. Justify your answer in each case. (a) Relation: x y if |x – y| < 1 Set: B = R B = Mn(R), the set of n xn matrices with real entries, where n is some fixed positive integer X ~ Y if Tr(X – Y) e Z (b) Set: Relation: (Here, Tr(X) denotes the trace of a matrix X € Mn (R).)

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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Let R be a relation on {a, b, c, d} defined as R = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, d)}. Show that the relation R is an equivalence relation using relation matrix.
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Which of these relations on {0, 1, 2, 3} are ?equivalence relations a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3,3)}
d