4. Show that the relation defined on Z as follows are equivalence relation (a) For all, n € Z, m R n <=> 3|(m² – n²). (b) Let A = : Z+ × Z+, define a binary relation R on A: (a, b) R (c,d) <=> a+d=c+ b for all (a, b) and (c,d) in A.
4. Show that the relation defined on Z as follows are equivalence relation (a) For all, n € Z, m R n <=> 3|(m² – n²). (b) Let A = : Z+ × Z+, define a binary relation R on A: (a, b) R (c,d) <=> a+d=c+ b for all (a, b) and (c,d) in A.
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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solve these two questions
![14. Show that the relation defined on Z as follows are equivalence relation
(a) For all, n e Z, m R n <=> 3|(m² — n²).
(b) Let A = Z+ × Z+, define a binary relation R on A:
(a, b) R (c,d) <=> a+d=c+ b for all (a, b) and (c,d) in A.
15. Prove that
(a) If A is a set, R is an equivalence relation on A, then distinct equivalence classes of
R form a partition of A; that is, the union of the equivalence classes is all of A, and
the intersection of any two distinct classes is empty.
..and a and b are elements of A,
(b) If A is a set, R is an equivalent relation on A, and a and b are elements of A, Then
either [a] n [b] = Ø or [a] = [b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9656a833-2928-468d-bda4-7d65920b0a10%2F353318bc-31f7-4ad8-877b-13900bd2cb0a%2Ffupeh3o_processed.png&w=3840&q=75)
Transcribed Image Text:14. Show that the relation defined on Z as follows are equivalence relation
(a) For all, n e Z, m R n <=> 3|(m² — n²).
(b) Let A = Z+ × Z+, define a binary relation R on A:
(a, b) R (c,d) <=> a+d=c+ b for all (a, b) and (c,d) in A.
15. Prove that
(a) If A is a set, R is an equivalence relation on A, then distinct equivalence classes of
R form a partition of A; that is, the union of the equivalence classes is all of A, and
the intersection of any two distinct classes is empty.
..and a and b are elements of A,
(b) If A is a set, R is an equivalent relation on A, and a and b are elements of A, Then
either [a] n [b] = Ø or [a] = [b].
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