Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 1
1. Let m, n ∈ N, and let S = {1, 2, . . . , m} and T = {1, 2, . . . , n}. How many
mappings are there from S to T ? How many of these are injective, and how many
are bijective?
2. Let m ∈ N. How many surjective mappings are there from {1, 2, . . . , m} to
{1, 2, 3}? [Hint: how many mappings are there? How many of these are not
surjective?]
3. Let S, T be sets and let α : S → T . Prove that α is onto if and only if there is a
function β : T → S such that α ◦ β = ιT .
4. Let S be a set, and let Sym(S) be the set of all bijections α : S → S.
Prove that Sym(S) is closed under composition.
[This means: prove that if α, β ∈ Sym(S), then α ◦ β ∈ Sym(S).]
5. Let S, T be sets. If α : S → T , let
Γ(α) = {(x, y) ∈ S × T : x ∈ S, y = α(x)}.
Γ(α) is called the graph of α.
(a) Prove that if α and β are mappings S → T , then α = β if and only if
Γ(α) = Γ(β).
(b) Let G ⊆ S × T . Prove that G is the graph of some mapping α : S → T if
and only if for every x ∈ S, there is exactly one element y ∈ T such that
(x, y) ∈ G.