How Can a Set Be Empty?
Date: 09/29/2003 at 01:27:50 From: Hash Subject: Why is empty set called a set ? Why is the empty or null set called a set when it has no elements? Is there a mathematical proof that it's a set?
Date: 09/29/2003 at 08:33:38 From: Doctor Peterson Subject: Re: Why is empty set called a set ? Hi, Hash. We try to make our definitions so that they are as useful as possible. In this case, we would like all the operations we can do between sets to yield sets, just as we want addition and multiplication of two numbers to produce a number. Now, what happens when you take the intersection of a pair of disjoint sets (sets with no elements in common)? The result is an empty set, right? If we didn't call that a set, then in this (rather common) case, the result of the intersection operation would not be a set. This is typical of the way math is done. We make some natural definition (for example, thinking of a set as any collection of objects), and then work with it; eventually we find that we have to refine our definitions, or clarify the extreme cases, in order to make our new branch of mathematics work neatly. We can't "prove" that the empty set is a set, since we are defining it as such; but we do have to demonstrate that it is a useful and consistent definition that produces interesting mathematics. It does! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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