MATH1061 ASSIGNMENT 4: DUE 22 May 4pm Semester 1, 2023
In writing your solutions make sure you give full details.
1. (6 marks)
(a) (2 marks) How many distinct arrangements can be made from the letters of the word
WOOLLOONGABBA ?
(b) (2 marks) How many distinct arrangements can be made from the letters of the word
WOOLLOONGABBA that begin with W and end with N ?
(c) (2 marks) What is the probability of choosing an arrangement of letters of the word
WOOLLOONGABBA at random, which starts with A and ends in N?
2. (10 marks) Let the relation µ on Z be defined by
xµy if and only if x2 ≡ y2(mod 4).
(i) (1 mark) Is µ reflexive?
(ii) (2 marks) Is µ symmetric?
(iii) (2 marks) Is µ antisymmetric?
(iv) (2 marks) Is µ transitive?
(v) (2 marks) Is µ is an equivalence relation? If so determine the equivalence classes.
(vi) (1 marks) Is µ a partial order?
In all cases justify your answer.
3. (20 marks) Let S = {1, 2} and define ρ and µ be relations on S defined by
ρ = {(1, 1), (1, 2), (2, 1)}
µ = {(1, 1), (2, 2)}.
For each of ρ and µ determine if they are
(i) (2 marks) reflexive;
(ii) (4 marks) symmetric;
(iii) (4 marks) antisymmetric;
(iv) (4 marks) transitive;
(v) (4 marks) an equivalence relation and, if so, determine the equivalence classes;
(vi) (2 marks) a partial order.
In all cases justify your answer.
4. (12 marks) Let G1 be a group defined on the set Z4 with binary relation, +, addition modulo
4, and let G2 be a group defined on the set Z5 − {[0]} with binary relation, ·, multiplication
modulo 5.
(i) (4 marks) Prove that both, G1 and G2, are cyclic and write down generators for each
group.
(ii) (4 marks) For each of the groups G1 and G2, determine all cyclic subgroups.
(iii) (4 marks) Determine whether the groups G1 and G2 are isomorphic. Justify your answer.