En anglais

Which sets get a size

Mathematics · Calculus & Analysis · ages 22-23

Name ______________________ Date ____________

On X = {1, 2, 3}, how many sets are in the smallest sigma-algebra containing {1}?
Answer: ______________
Which collection is a sigma-algebra on every space X?
- The empty set and X
- All one-point sets
- All finite sets
- All proper subsets of X
Which of these must belong to every sigma-algebra on a space X?
- Every set with exactly one point
- Every finite set
- The whole space X
A sigma-algebra is closed under complements: with every set it also holds that set's complement.
Circle one: True False
How do you show that a sigma-algebra is closed under countable intersections?
- Intersect the sets two at a time forever
- Complement each set, unite them, complement back
- List the sets in increasing size order
The empty set belongs to every sigma-algebra. Is this true?
Circle one: True False
What separates an algebra of sets from a sigma-algebra?
- An algebra need not be closed under taking complements
- An algebra is closed under countable intersections, not unions
- An algebra may lack closure under countable unions
- An algebra may omit the whole space and the empty set
On X = {1, 2, 3}, is the family with the empty set, {1}, {2}, and X a sigma-algebra?
- No, because the complement of {1} is missing
- Yes, because it holds unions and complements
- No, because it holds too many sets
A classmate says closing under finite unions is enough for limit arguments. Which reply spots the error?
- Limit sets can need countably many steps, so finite rules fall short
- Finite unions are never allowed inside a sigma-algebra
- A countable union is only a finite union written twice
How many sigma-algebras exist on a two-element set {a, b}?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Which sets get a size W1-mt_cpvegazufk-s1

4 · It contains the empty set, {1}, its complement {2, 3}, and X, for four sets.
The empty set and X · The collection containing only the empty set and the whole space is closed under complements and countable unions for every X.
The whole space X · A sigma-algebra always contains the whole space. Single points and finite sets need not belong.
True · That is one of the two closing rules.
Complement each set, unite them, complement back · The De Morgan laws turn an intersection into complements plus a union.
True · A sigma-algebra contains the whole space, and the complement of the whole space is the empty set.
An algebra may lack closure under countable unions · An algebra is closed under complements and finite unions. A sigma-algebra also has closure under countable unions.
No, because the complement of {1} is missing · The complement of {1} is {2, 3}, which is not in the family.
Limit sets can need countably many steps, so finite rules fall short · Sets built by limits, like points in infinitely many sets, need countable unions.
2 · There are the trivial sigma-algebra {empty set, {a, b}} and the full power set, for a total of two.