Discrepancies in mathematical definitions.

Question

What are some examples where there is a discrepancy in the mathematical definition of a term?

For example :

$\bullet$ Isosceles triangle: "exactly two sides are equal" or does it say "minimum two sides are equal"?

$\bullet$ Binomial Coefficient: $ {n \choose r} a^r \cdot b^{n-r}$ or $ {n \choose r} a^{n-r} \cdot b^r$

Are there any other such examples?

There is no discrepancy. For example, since $a+b=b+a$ in the binomial theorem, we have two expressions, which are of course equal. — Dietrich Burde, Jun 16 '19 at 12:09
@DietrichBurde that's true, but in the edit, the same value of $r$ gives two different terms. — AryanSonwatikar, Jun 16 '19 at 12:12
Now the terms are indeed different, but the binomial coefficient is only $\binom{n}{r}$. So there is no discrepancy for the definition of a binomial coefficient. — Dietrich Burde, Jun 16 '19 at 12:17
It seems to me that by "discrepancy" you really just mean "ambiguity", in other words terms whose definitions are not entirely agreed upon. For example some people use $\mathbb N$ to include zero and some not. — Jack M, Jun 16 '19 at 12:18
Some use countable to include finite, some don’t; also, some say dihedral group $D_n$ has $n$ elements, some $2n$ — J. W. Tanner, Jun 16 '19 at 12:31
Some ambiguities come from the fact that names differ from a country to another ; for example, "arithmetico-geometric" sequence is used for the product of an arithmetic and a geometric sequence in the US, and mean a sequence given by a relationship $u_{n+1}=au_n+b$ four some constants $a$ and $b$ in some other countries, like France. — Jean Marie, Jun 16 '19 at 12:49
A comma is used to separate thousands in US but as a decimal point in Europe. — J. W. Tanner, Jun 16 '19 at 13:36
according to some authors, a ring must have a multiplicative identity, but not according to others — J. W. Tanner, Jul 09 '19 at 00:35
in some contexts, a linear relationship is $y=mx+b$; in others, $y=mx$ — J. W. Tanner, Nov 01 '19 at 04:38
some use the symbol $\subset$ to mean subset; others use that symbol to mean proper subset — J. W. Tanner, Nov 26 '19 at 00:52
some multiply permutations right to left; others, left to right — J. W. Tanner, Nov 27 '19 at 15:36
for the complex inner product, some take the first argument to be conjugate-linear and the second to be linear, while others do the reverse — J. W. Tanner, Dec 22 '19 at 17:26
some use "range" to mean image; others use "range" to mean codomain — J. W. Tanner, Aug 06 '20 at 21:26

J. W. Tanner · Answer 1 · 2019-06-17T00:08:45.950

3

There are notations and terms in mathematics that have different conventions. For example, to some $\Bbb N$ is {$1,2,3,...$}, whereas to others $\Bbb N$ is {$0,1,2,3,...$}. For another example, some say a set is countable if there is an injective function from it to $\mathbb N$, whereas others say it has to be bijective. For another example, to some the dihedral group $D_n$ has $n$ elements, and to others it has $2n.$ In all of these cases, a writer using these should indicate which convention is being followed.

edited Jun 17 '19 at 00:08

answered Jun 16 '19 at 13:35

J. W. Tanner

60,406

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I would phrase the ambiguity with countably as "Some say a set is countable if it an injective function exists from it to $\mathbb{N}$ others say it has to be bijective. – Q the Platypus Jun 16 '19 at 13:40
Sincere question for @QthePlatypus: Why would you phrase it differently from the way I did? – J. W. Tanner Jun 16 '19 at 23:23
The way you phrase it sounds like the finite sets are arbitrarily thrown into the class of countable. But saying it has an injection into the naturals gives a single rule and suggests how you would prove that something is uncountable. – Q the Platypus Jun 16 '19 at 23:53
I guess I was trying to be less technical by saying some say a set is countable if it has the same cardinality as $\Bbb N,$ whereas to others it also includes the case where the set is finite, but I have edited based on the feedback – J. W. Tanner Jun 17 '19 at 00:10

Discrepancies in mathematical definitions.

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