## Symmetric difference is associative

I’m sure that any mathematics university student came across in the first years of university, in some course of set theory, to proving the fact that the difference symmetric operation is associative:

where .

## 167 sets

The following 167 sets satusfy the following conditions:

i)

ii) for all and .

Calculate .

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## Subsets

Let be subsets having elements of a set having elements. If the union of any three subsets in is different from , prove that the union of all the subsets in is different from .

## Sets

Let be a set having elements. Find the least possible value for such that there exist different subsets , having elements each, such that for any two elements of there exists such that contains one of them and does not contain the second one.

## Infinite cardinal numbers.

Prove that if is an infinite cardinal number, then and .

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