H 6. Prove that the distinct equivalence classes of the relation of congruence modulo n are the sets [0], [1], [2], ..., where for each a = 0, 1, 2, ..., n - 1, [a] = {m e Z | m = a (mod n)}. 1

Need clearly written understandable explanation so I may understand how do attached complex analysis question. Thankyou!!

8. Recall that for x, y = Z, x=y (mod 5) provided that x - y = 5t for some te Z. Describe the equivalence class of 0 under this equivalence relation.

Let R be a congruence relation modulo 7 on Z. Then the equivalence class [110] equals to which of the following: O [5] O 14] O [3] O [1] O [2] O o o o o

1)Consider the equivalence relation {(x,y)|x≡y(mod5)} on the set {n∈Z|−10⩽n⩽5}. Use the roster method to describe the following equivalence classes.   [−10]= [−9]=   2)The partially ordered set (Z+,≼) is the set of positive integers with the following partial order:a≼b if and only if a=b or a∣b (i.e., a divides b). Using the partial ordering ≼ on Z+, we find that: A. 56≺86 B. 56≻86 C. 56 and 86 are incomparable D. None of the above Using the partial ordering ≼ on Z+, we find that: A. 72≻12 B. 72≺12 C. 72 and 12 are incomparable D. None of the above Using the partial ordering ≼ on Z+, we find that: A. 35≻47 B. 35≺47 C. 35 and 47 are incomparable D. None of the above Using the partial ordering ≼ on Z+, we find that: A. 17≻51 B. 17≺51 C. 17 and 51 are incomparable D. None of the above   3) Consider the set A={1,4,7,8}. The partially ordered set (P(A),⊆) is the power set of A with elements partially ordered by inclusion. Which of the following sets are less…

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:

Define a relation R on the integers Z saying that (m, n) is in R if m2 is equivalent to n2 (mod 7). Show that R is an equivalence relation, and identify the equivalence classes of R.