# Question 1:Suppose that X is infinite. Show that the Fr ́echet filter on X is not an ultrafilter.    Question 2: Prove this statmentsLet X be a nonempty set. Given U, V, W ⊆ X × X , the following hold:1. (V−1)−1 =V.2. U ◦ (V ◦ W ) = (U ◦ V ) ◦ W .3. (U ∩ V )−1 = U−1 ∩ V −1.4. (U ◦ V )−1 = V −1 ◦ U−1. 5. U ⊆ V → U ◦ U ⊆ V ◦ V .

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Question 1:

Suppose that X is infinite. Show that the Fr ́echet filter on X is not an ultrafilter.

Question 2: Prove this statments

Let X be a nonempty set. Given U, V, W ⊆ X × X , the following hold:
1. (V−1)−1 =V.
2. U ◦ (V ◦ W ) = (U ◦ V ) ◦ W .

3. (U ∩ V )−1 = U−1 ∩ V −1.

4. (U ◦ V )−1 = V −1 ◦ U−1. 5. U ⊆ V → U ◦ U ⊆ V ◦ V .

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Step 1

As per norms, Questions 1  is answered (multiple subparts).Question 2 may be posted separately, as it deals with an independent notion of relations.  Question 1  relates to special  families of subsets of a set X, called filters and ultrafilters

Step 2

Definition of a filter (of subsets of X)

Step 3

Definition of an ultr...

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