Mathematics 1214: Introduction to Group Theory
Homework exercise sheet 1
Due 12:50pm, Friday 29th January 2010
1. List all of the mappings α : {1, 2} → {a, b}. Which of these are injective, which
are surjective and which are bijective?
2. Consider the mapping α : N → N, n 7→ n2 where N = {1, 2, 3, . . . } is the set of
positive integers.
(a) Is α injective? Is α surjective?
(b) Construct a function β : N → N such that β◦α = ιN , and check that α◦β 6= ιN .
3. Give examples of mappings R → R which are
(a) bijective;
(b) injective but not surjective;
(c) surjective but not injective;
(d) neither injective nor surjective.
Be sure to explain why your answers are correct.
4. Let m ∈ N. How many surjective mappings are there from {1, 2, . . . , m} to {1, 2}?
5. Let S, T be sets with S 6= ∅ and let α : S → T . Prove that α is one-to-one if and
only if there is a function β : T → S such that β ◦ α = ιS .
6. An operation ∗ on a set S is said to be commutative if a ∗ b = b ∗ a for all a, b ∈ S.
Complete the following Cayley table so that the operation • on {1, 5, 6} is commutative:
•
1
5
6
1
1
6
5
6
6
5
How many different ways are there of doing this?
7. Let M (2, R) be the set of 2×2 matrices with real entries, and let ∗ be the operation
on M (2, R) defined by A ∗ B = AB − BA for A, B ∈ M (2, R).
(a) Find three matrices A, B, C ∈ M (2, R) such that A ∗ (B ∗ C) = (A ∗ B) ∗ C.
(b) Find three matrices A, B, C ∈ M (2, R) such that A ∗ (B ∗ C) 6= (A ∗ B) ∗ C.