Difference Between Zero and NothingDate: 12/12/96 at 10:49:18 From: Alistair Cockburn Subject: The difference between 0 and nothing My four-year old asked, "What is the difference between 0 and nothing?" and my tongue got all tied up. How would you answer? Thanks - Alistair Cockburn Date: 12/12/96 at 23:05:40 From: Doctor Pete Subject: Re: The difference between 0 and nothing Well, first, I'm kind of surprised that a four-year old would even assume that there is indeed a difference between "zero" and "nothing," for it is not even clear to many older people (as you saw yourself) that these things are actually different concepts. Most mathematicians consider 0 to be a number, and "nothing" to be the empty set; they are related in that the empty set has zero elements in it; that is, the *cardinality* of the empty set is zero. To explain a bit more in detail, I will give a bit of set theory at this point. We can think of "sets" as collections of objects. For instance, we can have a set like: S = {dog, cat, horse, car} I've used the braces "{ }" to group the objects together. Each object (dog, for instance), is called an *element*. Such a collection consists of "subcollections," or *subsets*. That is, there is a subset of the above set which consists of those elements which are animals. Mathematicians say: A = { x in S : x is an animal } = {dog, cat, horse} We read this as, "A is the set of all x in S such that x is an animal." So we say that A is *contained* in S. Similarly, we can define another subset of S as: N = { x in S : x is a machine } = {car} Or we could have said: N = { x in S : x is not an animal } = { x in S : x not in A } = S \ A Here the backslash "\" is another notation mathematicians use, which is kind of like subtraction. What happens is we let N consist of elements in S which are not in A. Naturally, one might ask, what is E = { x in S : x is neither an animal nor a machine } ? Or, if we really want to be crazy, what is E = S \ S ? Well, it doesn't have any elements. Such a set is called the empty set, which is written as "{}" or a zero with a slash through it. Why this is not the same as 0 will become clear if we consider sets of numbers, rather than sets of objects. For example, let S = {0, 1, 2, 3, 4} What is the *cardinality* of S? That is, how many elements does S have? Clearly, it has 5. Mathematicians write this as |S| = 5. Now, consider the subset {0} of S. It contains a single element, 0. But it is not the empty set, for the empty set has *no* elements. Is the empty set a subset of S? Sure! To see why, ask yourself, "Is S a subset of itself?" Yes, because S contains itself, or every element of S is also an element of S (of course). Then S \ S must also be a subset of S. But this is, of course, the empty set. So both {} and {0} are valid subsets of S, but they are not the same. To see an example of the difference between 0 and {}, we ask, "what is the value of x such that 5 + x = 3 + 2 ? Clearly, x = 0 is the answer. Now, what about "what is the value of x such that: x + 5 = 1, and x + 1 = 1 Obviously, there is no answer; that is, x = {}. Now, hopefully, things are a bit more clear. The idea of "nothing" stems from this notion of a collection. Like eggs in a basket. If you had no eggs (nothing in the basket), then this is analogous to the empty set. The *number* of eggs in the basket is zero. So we can think of "nothing" as a term describing the set itself, whereas "zero" is a term not describing a set, but an element. The confusion between the two is a result of the fact that the number of elements in the empty set is 0. There's a subtle difference in that, one that perhaps a 4-year old might have a problem understanding. But it's definitely worth trying to explain. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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