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(Pairwise) distinct

Is there a difference between 'pairwise distinct' and 'distinct'?

If not, why would one use the term 'pairwise distinct'?

Note by Ton De Moree
3 years, 11 months ago

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When talking about sets, since all the elements are assumed to be distinct (namely that \( \{ 1, 1\} = \{1\} \) as sets), hence the terminology doesn't matter.

However, when we are not talking about sets, the could be a subtle difference, depending on the individual's interpretation. A common case is when we're talking about ordered tuples. There are some people who interpret the statement "\(a, b, c \) are distinct elements" to mean that "\(a, b, c\) are not all the same element. This can happen when \( a = b \neq c \). For example, I have received claims that \( (9,9,8) \) is ordered tuple of distinct elements since \( 9 \neq 8 \). As such, to stress that I actually mean \( a \neq b, b \neq c, c \neq a \), I often (though not religiously) use "\(a, b, c\) are pairwise distinct elements". This removes the ambiguity.

Another case, is when we're talking about multi sets (which comes with complications). In this case, from the multi set \( \{x_1 = 9, x_2 = 9, x_ 3 =8 \} \), the pair \( (x_1 = 9, x_2 = 9) \) can be considered distinct elements (from the multi set) since \( 1 \neq 2 \). However, if we want to talk about two unequal elements, we would thus use " pairwise distinct" to mean \( x_i \neq x_j \), as opposed to \( i \neq j \). Calvin Lin Staff · 3 years, 11 months ago

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@Calvin Lin But all this multiset business means is that when you are deciding when things are equal, you are comparing their whole structure. If you are comparing two multisets, for instance, these are only equal if:

  1. their underlying sets are the same,

  2. each element of the underlying set occurs the same number of times.

The fact that a particular element of an underlying set can occur more than once is not the issue.

In your example, the multisets \(\{9,9,8\}\), \(\{9,9\}\) and \(\{9,8\}\) are all distinct. The middle one is distinct from the other two because it does not contain \(8\), while the outside two are distinct because the first contains two \(9\)s, while the third contains just one \(9\).

In a similar vein, you would not say that two groups were equal just because they contained an identity element. Nor would you say that they were equal if they had the same number of elements in them. You might be prepared to say that they were equal if they were isomorphic (in which case you would be considering the identity of the isomorphism classes of the groups), or else you might decide to require exact identity of the groups, including their group structure.

The bottom line is that equality is a binary relation (written \(a = b\)). How that binary relation is defined depends on the set of objects being tested, but testing for equality can only be done two things at a time, and so things are always distinct exactly when they are pairwise distinct. The words "distinct" and "different" are the same in this context, and if you said "three different objects", you would not expect that two of them might be the same; you would expect all of them to be the different. Mark Hennings · 3 years, 11 months ago

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@Mark Hennings I agree with what you are saying; there is no difference mathematically (at least to me).

However, when considered linguistically, there is a world of difference, When dealing with students of varying sophistication, I have to choose phrases carefully for the clarity of language. The word "distinct" could have different meanings to beginner problem solvers (esp those whose native language is not English). Some think that "distinct" means "not all the same", since that is how they have personally interpreted the phrase. For a person that is experienced with the context, he will be familiar with the connotations and implications of various phrases that are used, and can point out why "distinct" is not "not all the same". For this reason "pairwise distinct", which means "no two are the same", is a clearer term than "distinct".

There is also the possibility, that in some corner of the world, they have defined "distinct" to mean "not all the same". While I have not seen this in an accredited source, I have no way of tracing local traditions. I have had students tell me that "A square is not a rhombus", "The complex numbers do not include the reals", "0 is a positive number" and "When asked to choose an element of the set, I can choose to choose no elements". These statements arise because there is no standardized terminology for everyone. Some of them are caused by poor teaching and misinterpretation, but others are accepted conventions in their part of the world.

On a related note, I have recently learn that the word trapezium has extremely different meanings in the US and the UK. Calvin Lin Staff · 3 years, 11 months ago

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@Calvin Lin OK, you are using the phrase to remove any possibility of ambiguity.

In some cases, however, there is a standardized terminology. The complex numbers do contain the reals - otherwise they could not be either a field or algebraically complete.

Let's agree to table this discussion (in the American sense, not the English!). Mark Hennings · 3 years, 11 months ago

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@Mark Hennings I thank you both for this discussion. Mark's thoughts very much resemble my own and he voiced them quite nicely. I understand why the current wording is used as well, I hadn't thought of fallacies and inconsistancies in teachings around the world. I very much agree that the wording here needs to be disambiguous.

In this case, the addition of 'pairwise' had me doubting that it might actually mean something else from what I first thought ("they must've added the word for a reason"). My english is good, but not perfect after all!

Never knew the US had different trapeziums! Good to know when encountering foreign students. Ton De Moree · 3 years, 11 months ago

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No. Equality, or distinctness, can only be tested against two things at a time.

There is a difference between "pairwise coprime" and "coprime", though. Mark Hennings · 3 years, 11 months ago

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