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 .
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
Definition of a filter (of subsets of X)
Definition of an ultr...
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