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''For explanation of the symbols used in this article, refer to the Table Of Mathematical Symbols .'' BASIC DEFINITION The intersection of ''A'' and ''B'' is written "''A'' ∩ ''B''". Formally: : ''x'' is an element of ''A'' ∩ ''B'' If And Only If
For example, the intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. The number 9 is ''not'' contained in the intersection of the set of Prime Number s {2, 3, 5, 7, 11, …} and the set of odd numbers {1, 3, 5, 7, 9, 11, …}. If the intersection of two sets ''A'' and ''B'' is empty, that is they have no elements in common, then they are said to be disjoint, denoted: ''A'' ∩ ''B'' = Ø. For example the sets {1, 2} and {3, 4} are disjoint, written {1, 2} ∩ {3, 4} = Ø. More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an Associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C. ARBITRARY INTERSECTIONS The most general notion is the intersection of an arbitrary ''nonempty'' collection of sets. If M is a Nonempty set whose elements are themselves sets, then x is an element of the intersection of M If And Only If For Every element A of M, x is an element of A. In symbols: : This idea subsumes the above paragraphs, in that for example, A ∩B ∩C is the intersection of the collection {A,B,C}. The notation for this last concept can vary considerably. Set Theorists will sometimes write "∩M", while others will instead write "∩A∈M A". The latter notation can be generalized to "∩i∈I Ai", which refers to the intersection of the collection {Ai : i ∈ I}. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the Index Set I is the set of Natural Number s, you might see notation analogous to that of an Infinite Series : : When formatting is difficult, this can also be written "A1 ∩ A2 ∩ A3 ∩ ...", even though strictly speaking, A1 ∩ (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras .) Finally, let us note that whenever the symbol "∩" is placed ''before'' other symbols instead of ''between'' them, it should be of a larger size. (Eventually this will be available in HTML as the Character Entity ⋂, but until then, try <big>∩</big>.)NULLARY INTERSECTION Note that in the previous section we excluded the case where M was the Empty Set (∅). The reason is the follows. The intersection of the collection M is defined as the set (see Set-builder Notation ) : If M is empty there are no sets ''A'' in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be ''every possible x''. When M is empty the condition given above is an example of a Vacuous Truth . So the intersection of the empty family should be the "set of everything". The problem is, ''there is no such set''. Assuming such a set exists leads to a famous problem in Naive Set Theory known as Cantor's Paradox . For this reason the intersection of the empty set is left undefined. There is nothing that can be done about the problem, it is just a fact of life in mathematics. A partial fix for this problem can be found if we agree to restriction our attention to subsets of a fixed set ''U'' called the '' Universe ''. In this case the intersection of a family of subsets of ''U'' can be defined as : Now if ''M'' is empty there is no problem. The intersection is just the entire universe ''U'', which is a well-defined set by assumption. SEE ALSO > |
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