PMA3014 Set Theory 2021/22 Prof. Martin Mathieu
Sheet 3—first marked homework
Each question is worth 5 marks.
Exercise 11.
(i) Let S be a set of sets and let X be another set. Show that(⋃
S
)×X =⋃{S ×X | S ∈ S}.
(ii) For arbitrary sets A, B and C prove the identity
A ∩ (B△C) = (A ∩B)△(A ∩ C),
where △ denotes the symmetric difference.
Exercise 12. Let X be a non-empty set. A filter on X is a non-empty setF of subsets
of X which satisfies the following properties:
(i) ∅ /∈ F;
(ii) for all A,B ∈ F we have A ∩B ∈ F;
(iii) for all A ∈ F and all Y ⊆ X we have: A ⊆ Y =⇒ Y ∈ F.
Let X = N, the set of natural numbers. Let n≤ = {m ∈ N | n ≤ m} for each n ∈ N
be the set of all natural numbers which are greater than or equal to n ∈ N. Let
F =
{
Y ∈ P(X) | ∃n ∈ N : n≤ ⊆ Y }. Show that F is a filter on N.
Exercise 13. In this question X, Y and Z denote non-empty sets. Suppose that
f : Z → Y is a function from Z to Y . Define
φ : XY −→ XZ , φ(g) = g ◦ f (g ∈ XY ).
(i) Show that φ is a well-defined function.
(ii) Suppose that f is surjective. Prove that φ is injective.
(iii) Suppose that φ is injective. Under what conditions does it follow that f is surjec-
tive?

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