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You are not logged in. #1 20131110 12:30:55
Logic QuestionI'm having a little trouble understanding how to interpret some logic statements. Is it correct to say that reads as "There is at least one x such that P(x,y) is true for all y"? Furthermore, how do I interpret these statements? Can we simply interpret them as "there exists an x AND there exists a y" and similarly for the subsequent statement? Or is this wrong? #3 20131110 18:57:04
Re: Logic Question
That is correct. As an illustration, let us apply this to the definition of continuity: let f(x) be a realvalued function of the real variable x. Then f is continuous iff the following holds: where denotes the statement . You should also be aware that and do not commute: . In the first statement, the y depends on the particular x chosen; different choices of x may require different y. In the second statement, however, there is a unique y for all the x you choose. For example, in the definition of continuity above, let us reverse the order of and : This is no longer the definition of continuity, but of uniform continuity, which is a stronger condition than continuity.
Right again. And this time commutes with , as does with . For example, the definition of continuity above can be restated as follows: It’s the same thing. 134 books currently added on Goodreads #4 20131111 00:00:56
Re: Logic QuestionThanks a lot  beautifully explained, makes perfect sense now. 