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Let N = {0, 1,2, ...} and let R be a binary relation on the set Z = {(a, b) : a, b e N} defined by (a1, b1) R(a2, b2) if and only if 3ª1561 < 3025b2. Also, let S be a binary relation on the same set Z such that (a1, b1)S(a2, b2) if and only if the sum (a1 – a2) + (b1 – b2) is divisible by 4. (a) Which of R and S is an equivalence relation? Prove it. (b) Which of R and S is a partial order? Prove it. (c) For the partial order you identified above, is every pair of elements of Z comparable? (You don't have to prove it.)

Let N = {0, 1,2, ...} and let R be a binary relation on the set Z = {(a, b) : a, b e N} defined by (a1, b1) R(a2, b2) if and only if 3ª1561 < 3025b2. Also, let S be a binary relation on the same set Z such that (a1, b1)S(a2, b2) if and only if the sum (a1 – a2) + (b1 – b2) is divisible by 4. (a) Which of R and S is an equivalence relation? Prove it. (b) Which of R and S is a partial order? Prove it. (c) For the partial order you identified above, is every pair of elements of Z comparable? (You don't have to prove it.)

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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Let N = ...} and let R be a binary relation on the set Z = {0,1, 2, {(a, b) : a,b e N} defined by (a1, b1) R(a2, b2) if and only if 3º151 < 302562. Also, let S be a binary relation on the same set Z such that (a1, b1)S(a2, b2) if and only if the sum (a1 – a2) + (b1 – b2) is divisible by 4. (a) Which of R and S is an equivalence relation? Prove it. (b) Which ofR and S is a partial order? Prove it. (c) For the partial order you identified above, is every pair of elements of Z comparable? (You don't have to prove it.) (d) For the partial order you identified above, draw the Hasse diagram for it restricted to {(a, b) E Z : a < 3,6 < 2}.
Transcribed Image Text:Let N = ...} and let R be a binary relation on the set Z = {0,1, 2, {(a, b) : a,b e N} defined by (a1, b1) R(a2, b2) if and only if 3º151 < 302562. Also, let S be a binary relation on the same set Z such that (a1, b1)S(a2, b2) if and only if the sum (a1 – a2) + (b1 – b2) is divisible by 4. (a) Which of R and S is an equivalence relation? Prove it. (b) Which ofR and S is a partial order? Prove it. (c) For the partial order you identified above, is every pair of elements of Z comparable? (You don't have to prove it.) (d) For the partial order you identified above, draw the Hasse diagram for it restricted to {(a, b) E Z : a < 3,6 < 2}.
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