I apologize if the title doesn't make sense, I am not fluent in the language of mathematics.

Suppose I have some arbitrary constant $α>1.$

Knowing just this, I am trying to interpret the meaning of $(α,β)∈(0,1)^2$

My first point of confusion is that I am not sure whether I am meant to view $(α,β)$ and $(0,1)$ as sets or as ordered pairs. To my understanding, sets are denoted with braces {x,y} whereas ordered pairs are denoted with parenthesis (x,y). Does the phrase "the ordered pair $(α,β)$ is an element of the ordered pair $(0,1)^2$" make any coherent sense? I assumed this was a mistake of notation, and I am actually trying to find {α,β}∈{0,1}$^2$

Secondly, assuming they are actually sets, I am confused about the relationship. Let the set A = {α,β} and the set B = {0,1}, such that $A∈B^2$.

I interpreted this to mean "the set A containing the elements $α$ and $β$ is an element of the Cartesian square of the set B containing the elements 0 and 1, given by the Cartesian product $B^2 = B*B$ = {(0,0),(0,1),(1,0),(1,1)} such that {α,β}$∈${(0,0),(0,1),(1,0),(1,1)}

But then I got stuck, since I am not sure how the set A containing two undetermined variables as elements can itself be an element of a set of ordered pairs.

Where am I going wrong here?

  • $\begingroup$ $(\alpha,\beta)$ is an ordered pair; $(0,1)$ is the set of real number strictly between $0$ and $1$. $\endgroup$ Oct 23, 2020 at 2:18
  • $\begingroup$ $(0,1)^2$ is almost certainly the collection of all points $(x,y)$ with $0\lt x\lt 1$ and $0\lt y\lt 1$; this is the topological notation. And $(\alpha,\beta)$ is a point on the plane. So all it is telling you is that $(\alpha,\beta)$ is a point on the plane with $0\lt\alpha\lt 1$ and $0\lt \beta\lt 1$. $\endgroup$ Oct 23, 2020 at 2:31

1 Answer 1


As with all ambiguous notation and language (which is to say, all notation and language), the interpretation depends on context. That said, the statement $$(\alpha,\beta)\in(0,1)^2$$ is very likely to mean that the ordered pair $(\alpha,\beta)$ is an element of the Cartesian product $(0,1)\times (0,1)$ where $(0,1)$ denotes the interval $\{x\in\mathbb{R}:0<x<1\}$ (not an ordered pair!). So in other words, $(\alpha,\beta)\in(0,1)^2$ just means $0<\alpha<1$ and $0<\beta<1$.

Of course, with your stated assumption that $\alpha>1$, this will never be true. So perhaps you are misunderstanding something about this assumption, or there is a typo somewhere, or this statement is just intended to be a false one.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.