**Question 1:**

** **

(a) Show that, for all sets A, B and C,

(i) (A ∩ B)^{ c} = A^{c}∩B^{c}.

(ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

(iii) A - (B ∪ C) = (A - B) ∩ (A - C).

(b) Let f: X → Y be a function and Ai ⊂ X for i ∈ I. Show that

(i) f(∩A_{i}) _∪f(A_{i}).

(ii) f(∪A_{i}) = ∪f(A_{i}).

**Question 2:**

** **

(a) Prove that the set G = {a + b√3 : a; b ∈ Z} forms an abelian group under ordinary addition.

(b) Let R be a relation in the set of all natural numbers N defined by " xRy if and only if x - y is divisible by 3." Show that R is an equivalence relation. Hence determine the equivalence classes of R.

**Question 3:**

** **

(a) A set G consists of ordered pairs (a, b) such that a, b are real numbers and a ≠ 0. An operation * is defined in G as follows

(a, b) * (c, d) = (ac, ad + b) where a, b, c, d are real numbers. Prove that the given set constitutes a non-abelian group for the given operation.

(b) Let (G,*) be a group. Prove the following:

(i) the identity element e is unique,

(ii) (a * b)^{-1} = b^{-1} * a^{-1} for a, b ∈ G.

**Question 4:**

** **

(a) Prove that a non-empty subset H is a subgroup of (G,*) if and only if

(1) a, b ∈ H ) a* b ∈ H.

(2) a ∈ H ) a^{-1} ∈ H where a^{-1} is the inverse of a in G.

(b) Prove that the set Q^{+} of all positive rational numbers forms an abelian group for the operation * defined by a * b = ab/ 2.

**Question 5:**

** **

(a) (i) Define a cyclic group.

(ii) Prove that any cyclic group is abelian.

(iii) The set G = {1,-1, I,-i} forms a cyclic group under multiplication of complex numbers.

- State a generator of G and show how it generates each element of G.
- Draw the Cayley table for G.

(b) Let H and K be two subgroups of a group G. Show that H ∩ K is a subgroup of G.