Exercise 4.4. Fix an integer m 2. Consider the relation R = {(x, y) E Z x Z : r = y (mod m)} (A) Prove that R is an equivalence relation (B) Determine the equivalence classes of R. (C) Describe the quotient set Z/R
Exercise 4.4. Fix an integer m 2. Consider the relation R = {(x, y) E Z x Z : r = y (mod m)} (A) Prove that R is an equivalence relation (B) Determine the equivalence classes of R. (C) Describe the quotient set Z/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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Transcribed Image Text:Exercise 4.4. Fix an integer m 2. Consider the relation
R = {(x, y) E Z x Z : r =y (mod m)}
(A) Prove that R is an equivalence relation
(B) Determine the equivalence classes of R.
(C) Describe the quotient set Z/R
Exercise 4.5. Let a1a2...an be an n digit number. Prove that
(A) a1a2 . .. an = a1 + a2 + · · ·+ an (mod 3)
(B) aja2 ... An = an (mod 5)
(C) aja2 . ..an = a1 + a2 + +an (mod 9)
+(-1)"+la1 (mod 11)
(D) a¡a2 -.. an = an an-1 + an-2-
Exercise 4.6. Compute the following quantities:
(A) 123 DIV 10
(B) 123 MOD 10
(C) 2178 DIV 9
(D) 2178 MOD 9
ho following greatest common divisors:
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