Duplicate Elements in Mathematical SetsDate: 08/31/2004 at 11:49:20 From: Derek Subject: Duplicate Elements in Mathematical Sets Can there be duplicate elements in a mathematical set? I've read definitions which suggest that a set of [red, blue, green] is the same set as [red, green, red, blue, blue, red]. Which suggests that duplications do not matter. Computer programming language sets tend to be defined in such a way that duplicates are not allowed. So just what IS the true definition? In event that duplicate elements are not allowed in a set, then are 1/2 and 2/4 considered duplicate elements? In my mind, they are uniquely indentifiable as being distinct elements, much like the words "black" and "noir." Technically speaking, a person who has no math background (!) would state that 1/2 and 2/4 are obviously different. Much as an alien to our world with no understanding of our language would never even think to equate "black" and "noir" even though they in fact do refer to the same colour. The intersection of the sets [1,2] and [2,3] is [2], correct? But if the sets were [1/2,1/3] and [2/4,3/9] then would the intersection be an empty set or... what? Date: 08/31/2004 at 14:22:40 From: Doctor Mike Subject: Re: Duplicate Elements in Mathematical Sets Hi Derek - Thanks for writing to Dr. Math. You've asked a variety of questions which are all related, and hopefully when I've answered each of them, the whole idea will make sense. First, can there be duplicate elements in a mathematical set? The short answer is no, but this will be answered further below. The important thing is what is IN the set, not how you DESCRIBE the set. Here is a simple illustration. Suppose a room contains ten adult males, ten adult females, ten non-adult males, and ten non-adult females, for a total of 40 people. Let's define the set S to be the set of all people in the room who are either "female" or "non-adult". What is the size of that set? I have carefully worded the definition to exclude the adult males, so those ten people are disqualified for membership in the set, leaving 30. If you reason that the ten non-adult females must be counted twice, then the "head count" in the set S goes like this: 10 for the 10 adult females, because they are female 10 for the 10 non-adult females, because they are female 10 for the 10 non-adult females, because they are non-adult 10 for the 10 non-adult males, because they are non-adult ---- 40 TOTAL You can't exclude 10 of the 40 and still have 40 left. That does not make sense, which is why set membership is defined that way. The idea behind this is that there can be many different ways to QUALIFY for membership in the set, but a particular thing is either in the set, or not in the set. Next, is the set [red, blue, green] the same set as [red, green, red, blue, blue, red]? The 2 sets of colors you have given both have 3 elements, and they are the same elements, thus they are the same sets. The word "sequence" is used for when repetition may be important. If your examples were considered as sequences, the first would be a 3-element sequence, and the second a 6-element sequence, and they would be different. Third, would 1/2 and 2/4 or "black" and "noir" be considered duplicate elements? The apparent problem with your black/noir example is eliminated when you examine what exactly is the nature of the elements in the set. Are the elements "colors" or "words"? If the set is a set of colors, then the set has one element. An English speaker would say that "black is in the set" and a French speaker would say that "noir est dans l'ensemble". If the set is a set of words, then the set has two elements. "Black" and "noir" are different words, even if their meaning is the same. For your question about 1/2 and 2/4, we can clarify it in a similar way. If you intend these things as rational numbers, then they mean the same thing. Once you know that 1/2 is in the set, then you also know that 2/4 and 1001/2002 and many others are in the set as well. However, if by 1/2 and 2/4 you mean the character strings "1/2" and "2/4" which have 3 characters each, that says nothing about the character string "1001/2002" with 9 characters. I should comment on your use of the word "allowed". It is true that duplicate elements ARE ALLOWED when you specify a set, but duplicates do not enlarge the set at all. Finally, the intersection of {1/2,1/3} and {2/4,3/9) is the intersection of 2 identical sets. For any set A, the intersection of A and A is A. I hope this clears it up for you. It is appropriate to make sure you know what the definition is, so you can use the concept of set correctly. I've also mentioned the related word "sequence". If you have further questions, please feel free to write back. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/ |
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