relation on the integers:

Question 1. (a) Define the following relation on the integers:

a ~ 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 equivalence relation. How many distinct equivalence classes are. there? List them.

Question 2. (a) Let A and B he two sets with A g B and IAI r IBI (T ’30. prove that A .- B.

(h) Give an example showing that part (a) is false if lAl = IBI = That is. give an example of two sets A

and B with A g B. IAI = [Bi = 00. hut A yé B.

Question 3. Let. n he a natural number with n 2 2. Define [a] z {b e Z : a -:- b(mod 17.)} and

Zn z : I 6 As mentioned in class: 5 is an equivalence relation on the integers.

(a) Prove that Z” = l]. [n – Hint. Use the division algorithm and the fact that equivalence

ole pnrtition a set (these were. provan in class).

(h) Define [a] + .- [a + b]. If = [z] = [1r]. prove that + [z] = [y] + [w].

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

Question 4. Let n be a natural number with n 2 2. Define U(n) = : :r e Z. god(13.n) = 1}.

(a) Define. [a][b] = lab]. If [(2] = = Show that

(h) Prove. that U(n) is a gmup under the operation defined in part (a).

Question 5. Show that the {5. 15, 25. 35} is a group under multiplication modulo <10.

Question 6. (a) Supposic. G is a finite group- Prove that every element in G has finite order.

(b) Let G be a group (not necessarily finite). Prove that lgl = lg ’ ‘l for 8.11 g E C.