Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 144 / 2 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

A filter on a set

U

on

N

is a nonempty family of subsets of

N

such that:

$\emptyset \in / U$ ;
if $A, B \in U$ , then $A \cap B \in U$ ;
if $A \in U$ and $A \subseteq B \subseteq N$ , then $B \in U$ .

These conditions imply

N \in U

.

Solved by gpt-5.6-sol high.

ii

Solution

An ultrafilter is a proper filter maximal under inclusion. Equivalently, for every

A \subseteq N

, exactly one of

A

and

N ∖ A

belongs to

U

.

Solved by gpt-5.6-sol high.

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