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Is the intersection of two infinite sets always an infinite set?
No. It can be infinite, finite or null. The set of odd integers is infinite, the set of even integers is infinite. Their intersection is void, or the null set.
What is a disjoint set?
ExplanationFormally, two sets A and B are disjoint if their intersection is the empty set, i.e. if This definition extends to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint if, given any two sets in the collection, those two sets are disjoint.Formally, let I be an index set, and for each i in I, let Ai be a set. Then the family of sets {Ai : i ∈ I} is pairwise disjoint if for any i and j in I with i ≠ j,For example, the collection of sets { {1}, {2}, {3}, ... } is pairwise disjoint. If {Ai} is a pairwise disjoint collection (containing at least two sets), then clearly its intersection is empty:However, the converse is not true: the intersection of the collection {{1, 2}, {2, 3}, {3, 1}} is empty, but the collection is not pairwise disjoint. In fact, there are no two disjoint sets in this collection.A partition of a set X is any collection of non-empty subsets {Ai : i ∈ I} of X such that {Ai} are pairwise disjoint andSets that are not the same.
Difference between Union and Set Intersection Operation?
union means to group the given sets. where as intersection means to pick out the common elements from the given sets. if set a has 1,2,3 elements and B has 1,2,3,4,5. then its union will have 1,2,3,4,5 as its elements. and its intersection will have 1,2,3 as its elements.
Difference betweenDisjoint sets and pairwise disjoint sets?
Assuming that, by 'disjoint', you mean that a collection of sets has an empty intersection, here is the difference between pairwise disjoint and 'disjoint': If a collection of sets is pairwise disjoint, it implies that the collection is 'disjoint': If no two sets overlap, then no k sets would overlap for any k, since this would require the overlap of at least two sets i.e. you know for sure that k things aren't in contact at a common point if you know that no two of them are in contact with each other. However, if a collection of sets is 'disjoint' (so the overall intersection is empty), it doesn't mean that the collection is pairwise disjoint. For instance, you could have a collection of 4 sets containing two overlapping pairs, where no set in one pair overlaps with a set in the other. So the intersection of the whole thing would be empty without pairwise disjointness. You could have a few things in contact with each other without all of them sharing a point of contact.
Does the set of all sets other than the empty set include the empty set?
The collection of all sets minus the empty set is not a set (it is too big to be a set) but instead a proper class. See Russell's paradox for why it would be problematic to consider this a set. According to axioms of standard ZFC set theory, not every intuitive "collection" of sets is a set; we must proceed carefully when reasoning about what is a set according to ZFC.
