Mth644 Assignment 1 2023 |100% Correct Solution

 

Mth644 Assignment 1 2023 |100% Correct Solution

 Question: 1

Let X = R and H be the collection of all countable subsets of X. Show that H is

o -ring but not an algebra of sets.

Marks: 10

Solution:

To show that the collection H of all countable subsets of X = R is a o-ring but not an

algebra of sets, we need to verify the following properties:

b)Forany Sequence (A 1° of subsets of"y* I

Let's check these properties:

a) Closure under relative complements (A- B€H):

Suppose A and B are countable sets in H, i.e., A and B are countable subsets of R.

We want to show that their relative complement, A \ B, is also countable.

The relative complement A \ B contains all elements in A that are not in B. Since A

and B are countable, the set A \ B can be seen as a subset of A, which is countable.

Hence, A \ B is also countable.

Therefore, H is closed under relative complements.

b) Closure under countable unions (U {A4є H):

Suppose {Ag) is a countable collection of subsets in H. Each A, is countable. We want

to show that their union, U = U A;, is also countable.

Let's define the set B = {x € R | x is in at least one of the sets An}. Since each An is

countable, B is the countable union of countable sets, and thus B is countable.

Now since This a subset of B Ilis also countable since it is a subset of a countsble set.

Therefore the countable union U = U°countable unions

= 1An is countable, and H is closed under.

we have shown that H is closed under countable unions and relative complements, which

are the defining properties of a o-ring.

Now, to show that H is not an algebra of sets, we need to find a counterexample. An

algebra of sets must satisfy additional conditions compared to a o-ring,