Mathematics

Question 1. (a) Define the following relation on the integers:
(1 ~ b ⇔ 5I(2a + 3b)

Prove that this is an equivalence relation. Find the equivalence classes and
(b) Define the following equivalence relations on the integers:

a ↽∼⋅ b ⇔ a + b is even
Prove that this is an cquimlcncc relation. How many distinct equivalence classes are. there? List them.
Question 2. (a) Lot. .4 and B be two sets with A ⊆ B and [Al 2 [Bl (f ’30. prove that A ≓∙ B.
(b) Give an emnple showing that pan (:1) is false if IAI = IBI = ac. That is. give an example of two sets A
and B with A ⊆ B. IAI = [Bi = 00. but A ⋥≙ B.

Question 3. Let n be a natural munbex with n ≥ 2. Define [a] z {b E Z : a ≡ b(mod 11)} and
Zn ∶ ∶ J’ ∈ Z}. As mentioned in class: ≡ is an equivalence rplation on the integers.

(a) Prove that Z” = [n – Hint. Use the division algorithm and the fact that equivalence
clamps partition « set (them were prowl] in clam).

(b) Define [a] + =- [a + b]. If = [z] 2 {Ir}. prove that + [z] = [y] + [w].

(c) Show that Z” is a group under the operation defined in part (1)).

Question 4. Let n be a natural number with n ≥ 2. Define U(n) = : .r ∊ Z, god(r.n) = 1}.

(a) Define [aflb] = {all}. If [1:] = = Show that MU]

(b) Prom that Uo1) is a group under tho operation defined in part (a).

Question 5. Show that the {5. 15, 25. is a group under multiplication modulo 40.

Question 6. (a) Supposw. G is a finite group- Prove that eve-1y element in G has finite order.
(b) Let C be a group (not necessarily finite). Prove than lgl = Ig’ ∣∣ for all g ∈ 0.
Question 7. Fix an integer n ≥ 2. Define
G” = {f : Z -> 2] f is bijmttive and f(i +1!) 2 f(r’) + :2 Vi ∈ Z}
Show that G” is a group with composition of maps as the binary operation (You do not have to prove t!
this is an associative operation. since map composition is always associative).

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