6 - are likely to construct will not encounter this...

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Approach (2) can be generalised to a general axiom of comprehension , which constructs sets by taking all elements of a set which satisFes some property P ( x ) . In fact, you have seen a similar construct in Haskell: the construction [x | x <- S, p x] denotes a list of terms x such that x is in the list S and the property p ( x ) holds for some predicate p . However, a completely unrestricted use of com- prehension can cause problems: Russel’s paradox: the construction R = { X : X is a set X ±∈ X } is not a set 1 . Assume R is a set. If R R , then by the construction of R it follows that R ±∈ R which is impossible. If R ±∈ R , then by the construction of R it follows that R R and again we have a contradiction. Hence, the assumption must be wrong and R cannot be a set. It is possible to remove this sort of paradox using axiomaticsettheory , which is a very formal deFnition of set theory. This deFnition is beyond the scope of this course, and any ‘normal’ sets you
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Unformatted text preview: are likely to construct will not encounter this problem. 2.2.1 Basic Set Constructors We describe set constructors which build new sets from old. In each case. the resulting sets can be assumed to be well-deFned as long as the original sets are well-deFned. D EFINITION 2.5 (C OMBINING SETS ) Let A and B be any sets. We may construct the following sets: Set Union A B = { x : x A x B } Set Intersection A B = { x : x A x B } Difference A-B = { x : x A x B } Symmetric difference A B = ( A-B ) ( B-A ) 1 A colloquial rendition of this paradox is: In a certain town, Kevin the barber shaves all those and only those who do not shave themselves. Who shaves the barber? The riddle has no good answer. 7...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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