Noting Nothing: Braces around Zero vs. Empty BracesDate: 08/02/2010 at 09:36:22 From: Ien Subject: difference between null set,phi and 0 in braces I'm confused: what is the difference between the null set, the "phi," and a zero enclosed in braces? I don't know which is which. Date: 08/02/2010 at 10:12:55 From: Doctor Ian Subject: Re: difference between null set,phi and 0 in braces Hi Ien, The null set is a set that contains no elements. We can represent that with empty brackets, like so, ... {} ... or with a phi character. Those two representations are equivalent, and either can be used in place of the other. But a zero enclosed in braces, like this, ... {0} ... is a set with one element -- namely, the number zero. It's similar to the set {5}, except it has 0 as an element, instead of 5. Note, too, that the set {phi} is a set that contains a set. For example, we can have a set like this: A = { {a,b}, {c,d,e}, {a,c} } Here, A is a set that contains some other sets as elements. Since this set can contain sets, it can contain the empty set, too: A = { {a,b}, {c,d,e}, {a,c}, {} } And we can remove elements from the set, one at a time: A = { {c,d,e}, {a,c}, {} } A = { {a,c}, {} } A = { {} } That last set has a single element. That element is the empty set. And since {} and phi are two ways of saying the same thing, we can write that as A = { phi } But this is very different than ... B = phi = {} ... which is the empty set itself -- not a set containing the empty set. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 08/04/2010 at 05:51:36 From: Ien Subject: Thank you (difference between null set,phi and 0 in braces ) Thanks for the answer. It helped me a lot. |
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