Confused About Set Notation: A set of two variables is an element of the Cartesian square/product of a set of two numerical values?

Question

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?

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.