Produce an infinite collection of sets a1, with the property that every ai has an infinite number pf elements , ai inter aj = empty set for all i, j different.

and uinfinitii=1ai =n

Step-by-step explanation:

For this case we need to produce an infinite collection of sets with the property that every has an infinite number of elements with and

So for this case we need to create a set A who satisfy 3 conditions (Infinite number of elements, Disjoint and Union represent the natural numbers)

So if we define the nth term for the set A:

We see that the set A represent all the odd multiplies of and if we check the properties we have this:

Disjoint

If we select with and we can assume for example that  and if we have an element a in the intersection of the sets , so then needs to exists some odd numbers k and l such that

And since we assume that then we have that and we can write:

And then and if we divide by we got:

so then k is not odd since the last statement contradicts this. So then we can conclude that

Union

For this case we need to show that  

Since each element is a subset of the natural numbers then the unision of the sets represent N

For the other side of the explanation if we assume that we can write for any and k odd, and by this and we chaek the property.

Infinite condition

For this case

is an infinite set since we don't have a limit for n so then we have infinite elements for this case.

And since all the properties are satisfied we end the problem.

step-by-step explanation:

the true ones are a b and d