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In mathematical writing, what are the differences between the words distinct and different?

  • The vertices $x$ and $y$ are in distinct components of the graph $G.$
  • The vertices $x$ and $y$ are in different components of the graph $G.$

Merriam-Webster Dictionary isn't being terribly helpful:

  • distinct - distinguishable to the eye or mind as being discrete (i.e., constituting a separate entity: individually distinct) or not the same: separate
  • different - not the same: distinct
ryang
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Max Flow
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    They are synomymous. In your example I think I'd go with different but I don't know why. Even if one adjective is microscopically "better" than the other, there are much more important things to worry about in mathematical writing. Like whether to call the vertices $x$ and $y$, or $u$ and $v$, or $v$ and $w$, and whether they are vertices or nodes or points, and whether the graph is $G$ or $\Gamma$. The difference (distinction?) between distinct and different is the least of your worries. – user14111 May 07 '25 at 08:25
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    The two words are distinct, but not really different. – Gerry Myerson May 07 '25 at 09:28
  • There may be a small difference in meaning: distinct stresses that things are strictly separated, but different merely indicates that they are not the same. E.g., "distinct sets" means there is no intersection, "different sets" means that at least one element is not in both sets. Other example: a mathematical concept/entity may have different definitions, but still the two entities may be the same (after it is shown that the definitions are equivalent.) – R. J. Mathar May 07 '25 at 09:48
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    @R.J.Mathar "distinct sets" does not mean "there is no intersection". It is different from "disjoint sets". – Anne Bauval May 08 '25 at 05:49
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    In addition to some of the nice answers here, I would add that "distinct" is a technical term (meaning "not equal") whereas "different" is just an English word. Two mathematicians might disagree about whether two mathematical objects are "different", but in a given context we should in principle all agree about whether two objects are "distinct". – Jim Belk May 08 '25 at 09:21
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    @R.J.Mathar I think that in English the word distinct is weaker not stronger than difference because there is something called a distinction without a difference but I never heard of a "difference without a distinction." – user14111 May 08 '25 at 18:39
  • @user14111 I also noticed R. J. Mathar's (common) abuse of language (whence my quotation marks), but wanted to insist on his main misunderstanding. – Anne Bauval May 08 '25 at 19:15

2 Answers2

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In my opinion they mean the same thing, but the emphasis is not quite the same. So while either choice will still be technically correct, often one will be more natural.

I think that "different" is more usual when the alternative would be "the same", whereas "distinct" is more usual when the alternative would be "not necessarily distinct" (which might not be explicit).

In your example, sometimes you will want to say "$x$ and $y$ are in different components", and sometimes you will want to say "$x$ and $y$ are in the same component", but you would not normally need to say "$x$ and $y$ are in (not necessarily distinct) components of $G$". (The exception would be if you are giving the components of $x$ and $y$ names, and simultaneously noting whether they are necessarily distinct.)

On the other hand, we might want to write "let $m$ and $n$ be distinct positive integers" or "let $m$ and $n$ be (not necessarily distinct) positive integers", but it would seem strange to write "let $m$ and $n$ be the same positive integer".

Especially Lime
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  1. The adjective "distinct" refers to identity and non-duplication—and is more specific than its synonym "different":

    • The numbers 1,2,3,3 are not distinct, but are arguably different.

  2. In the following, "different" is the more natural word choice, whereas "distinct" would sound odd, particularly in the first two sentences:

    • We will solve the PDE using two different techniques.
    • There are three different cases to check.
    • Consider two vertices in different components of the graph $G.$

  3. In the following, "distinct" is the more natural word choice, while using "different" would lose the emphasis on non-identicality:

    • The sets $\{1,2\}$ and $\{2,3\}$ are distinct (though not disjoint).
    • The polynomial has three distinct real roots.

  4. Gerry Myerson's witticism from the comments

    • The two words are distinct, but not really different

    drives home the point that "distinct" (being the more specific word) is generally replaceable with "different" whereas the converse isn't true: notice that it can be reworded as

    • The two words are different, but not really different,

    but not as

    • The two words are distinct, but not really distinct.
ryang
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